https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-2/v/introduction-to-limits-hd 2023-07-27T01:47:53.920337803Z Limits intro In this video, we learn about limits, a fundamental concept in calculus. Limits help us understand what a function approaches as the input gets closer to a certain value, even when the function is undefined at that point. The video demonstrates this concept using two examples with different functions. Sal Khan video https://cdn.kastatic.org/googleusercontent/J1zqnnl2yj2g_ZG6KjF-DQT1hCeNwc1bHWc0qmXwvoFoGSz6w8z729suyX8w6AS3EG4jKcQoDVs6suPJJqryyUSFXw Limits intro In this video, we learn about limits, a fundamental concept in calculus. Limits help us understand what a function approaches as the input gets closer to a certain value, even when the function is undefined at that point. The video demonstrates this concept using two examples with different functions. https://cdn.kastatic.org/ka-youtube-converted/riXcZT2ICjA.mp4/riXcZT2ICjA.mp4 692 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-5a/v/limit-properties 2023-07-27T01:47:53.920337803Z Limit properties This video introduces limit properties, which are intuitive rules that help simplify limit problems. The main properties covered are the sum, difference, product, quotient, and exponent rules. These properties allow you to break down complex limits into simpler components, making it easier to find the limit of a function. Sal Khan video https://cdn.kastatic.org/googleusercontent/BPiLl-iws3jpH0YefJaW5pCzEjpge04AROcRXk7U96G0qIYHERrdkQtVfE1yb7R1I7OBgNHqou4X-ha9rFrUiWsp Limit properties This video introduces limit properties, which are intuitive rules that help simplify limit problems. The main properties covered are the sum, difference, product, quotient, and exponent rules. These properties allow you to break down complex limits into simpler components, making it easier to find the limit of a function. https://cdn.kastatic.org/ka-youtube-converted/lSwsAFgWqR8.mp4/lSwsAFgWqR8.mp4 308 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-5a/v/limits-of-combined-functions 2023-07-27T01:47:53.920337803Z Limits of combined functions In this video, we learn how to find the limit of combined functions using algebraic properties of limits. The main ideas are that the limit of a product is the product of the limits, and that the limit of a quotient is the quotient of the limits, provided the denominator's limit isn't zero. video https://cdn.kastatic.org/googleusercontent/aNJcazKFieUpZiIPBRChPxyAYRNuDMFh7vIvy8t2yPdzcOgBLlMOpA0yrU0KWnHm5Ss_wECfzPI-tDL_DkqJvKs Limits of combined functions In this video, we learn how to find the limit of combined functions using algebraic properties of limits. The main ideas are that the limit of a product is the product of the limits, and that the limit of a quotient is the quotient of the limits, provided the denominator's limit isn't zero. https://cdn.kastatic.org/ka-youtube-converted/AhnJtTI_DMM.mp4/AhnJtTI_DMM.mp4 249 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-5a/v/limits-of-combined-functions-piecewise 2023-07-27T01:47:53.920337803Z Limits of combined functions: piecewise functions This video demonstrates that even when individual limits of functions f(x) and g(x) don't exist, the limit of their sum or product might still exist. By analyzing left and right-hand limits, we can determine if the limit of the combined functions exists and find its value. video https://cdn.kastatic.org/googleusercontent/--1wvYhRbN3bp_a9BUd90tIgY49yurPjDOxfeHCR49aSXp0sjP8Mrn6YtfqHGGmT3BpCdsVM9Gxj8Mwi_P3YvFDl Limits of combined functions: piecewise functions This video demonstrates that even when individual limits of functions f(x) and g(x) don't exist, the limit of their sum or product might still exist. By analyzing left and right-hand limits, we can determine if the limit of the combined functions exists and find its value. https://cdn.kastatic.org/ka-youtube-converted/Xo49w4DDEfQ.mp4/Xo49w4DDEfQ.mp4 253 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-5b/v/limit-by-substitution 2022-09-27T03:20:10.805796422Z Limits by direct substitution Sal explains how you can easily find limits of functions at points where the functions are continuous: simply plug in the x-value into the function! Later we will learn how to find limits even when the function isn't continuous. video https://cdn.kastatic.org/ka_thumbnails_cache/de57d77c-2ca0-46dc-8d5a-a313b891f121_1280_720_base.png Limits by direct substitution Sal explains how you can easily find limits of functions at points where the functions are continuous: simply plug in the x-value into the function! Later we will learn how to find limits even when the function isn't continuous. https://cdn.kastatic.org/ka-youtube-converted/cfOcOgr0E7U.mp4/cfOcOgr0E7U.mp4 127 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-5b/v/undefined-limit-by-substitution 2022-09-27T03:20:10.805796422Z Undefined limits by direct substitution Sal gives an example of a limit where direct substitution ends in a quotient with 0 in the denominator and non-0 in the numerator. Such limits are undefined. What about limits where substitution ends in 0/0? Keep going and you'll see! video https://cdn.kastatic.org/ka_thumbnails_cache/4a9590f4-bf01-48e5-a2a6-0f2e0397a886_1280_720_base.png Undefined limits by direct substitution Sal gives an example of a limit where direct substitution ends in a quotient with 0 in the denominator and non-0 in the numerator. Such limits are undefined. What about limits where substitution ends in 0/0? Keep going and you'll see! https://cdn.kastatic.org/ka-youtube-converted/JlOyb_RTejo.mp4/JlOyb_RTejo.mp4 134 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-6/v/limit-example-1 2023-10-20T17:49:30.470270187Z Limits by factoring In this video, we explore the limit of (x²+x-6)/(x-2) as x approaches 2. By factoring and simplifying the expression, we discover that the function is undefined at x = 2, but its limit from both sides as x approaches 2 is in fact 5. Sal Khan video https://cdn.kastatic.org/googleusercontent/D5DvW40fHhxKEOVQeAGWJsz4bfCOxL4VtwJY2M7b3X8pjYFCZfhoK5ges5woh73cAcug89AM9CA9xXZNY-8DfQKG Limits by factoring In this video, we explore the limit of (x²+x-6)/(x-2) as x approaches 2. By factoring and simplifying the expression, we discover that the function is undefined at x = 2, but its limit from both sides as x approaches 2 is in fact 5. https://cdn.kastatic.org/ka-youtube-converted/EAa3J_nDkoI.mp4/EAa3J_nDkoI.mp4 344 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-5b/v/limits-of-piecewise-functions 2023-07-27T01:47:53.920337803Z Limits of piecewise functions In this video, we explore limits of piecewise functions using algebraic properties of limits and direct substitution. We learn that to find one-sided and two-sided limits, we need to consider the function definition for the specific interval we're approaching and substitute the value of x accordingly. video https://cdn.kastatic.org/googleusercontent/MN8eQ6wHnZSXGRVLTSSXGf05-TPQ15QbYTU2fvwXRXySaSNxTYGX--GyeGctHs6wJRM1_WF5xoxq2zh8EJrD_Q0 Limits of piecewise functions In this video, we explore limits of piecewise functions using algebraic properties of limits and direct substitution. We learn that to find one-sided and two-sided limits, we need to consider the function definition for the specific interval we're approaching and substitute the value of x accordingly. https://cdn.kastatic.org/ka-youtube-converted/2xdh0yKopB8.mp4/2xdh0yKopB8.mp4 229 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-5b/v/limits-of-trigonometric-functions 2023-07-27T01:47:53.920337803Z Limits of trigonometric functions This video covers limits of trigonometric functions, focusing on sine, cosine, and tangent. It emphasizes that sine and cosine are continuous and defined for all real numbers, so their limits can be found using direct substitution. For tangent and cotangent, limits depend on whether the point is in their domain. video https://cdn.kastatic.org/ka_thumbnails_cache/44e567a3-14b4-4665-ac9e-0d6167f8a012_1280_720_base.png Limits of trigonometric functions This video covers limits of trigonometric functions, focusing on sine, cosine, and tangent. It emphasizes that sine and cosine are continuous and defined for all real numbers, so their limits can be found using direct substitution. For tangent and cotangent, limits depend on whether the point is in their domain. https://cdn.kastatic.org/ka-youtube-converted/5_zxtSyTfOY.mp4/5_zxtSyTfOY.mp4 366 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-8/v/squeeze-sandwich-theorem 2021-04-13T17:49:49Z Squeeze theorem intro The squeeze (or sandwich) theorem states that if f(x)≤g(x)≤h(x) for all numbers, and at some point x=k we have f(k)=h(k), then g(k) must also be equal to them. We can use the theorem to find tricky limits like sin(x)/x at x=0, by "squeezing" sin(x)/x between two nicer functions and ​using them to find the limit at x=0. Sal Khan video https://cdn.kastatic.org/googleusercontent/G0rQ_8hX_C-zwc7sPVnCxTD1hn3dyjwhQeD1n5HJfVc688cr5iCld193WSQJUznZFe7VClwzmPyGQv_lz5d5r5IQ Squeeze theorem intro The squeeze (or sandwich) theorem states that if f(x)≤g(x)≤h(x) for all numbers, and at some point x=k we have f(k)=h(k), then g(k) must also be equal to them. We can use the theorem to find tricky limits like sin(x)/x at x=0, by "squeezing" sin(x)/x between two nicer functions and ​using them to find the limit at x=0. https://cdn.kastatic.org/ka-youtube-converted/WvxKwRcHGHg.mp4/WvxKwRcHGHg.mp4 431 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-8/v/sinx-over-x-as-x-approaches-0 2023-07-27T01:47:53.920337803Z Limit of sin(x)/x as x approaches 0 In this video, we prove that the limit of sin(θ)/θ as θ approaches 0 is equal to 1. We use a geometric construction involving a unit circle, triangles, and trigonometric functions. By comparing the areas of these triangles and applying the squeeze theorem, we demonstrate that the limit is indeed 1. This proof helps clarify a fundamental concept in calculus. video https://cdn.kastatic.org/ka_thumbnails_cache/e2078fcb-1f39-4217-8053-927effb87a0a_1280_720_base.png Limit of sin(x)/x as x approaches 0 In this video, we prove that the limit of sin(θ)/θ as θ approaches 0 is equal to 1. We use a geometric construction involving a unit circle, triangles, and trigonometric functions. By comparing the areas of these triangles and applying the squeeze theorem, we demonstrate that the limit is indeed 1. This proof helps clarify a fundamental concept in calculus. https://cdn.kastatic.org/ka-youtube-converted/5xitzTutKqM.mp4/5xitzTutKqM.mp4 556 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-8/v/1-cosx-over-x-as-x-approaches-0 2023-07-27T01:47:53.920337803Z Limit of (1-cos(x))/x as x approaches 0 In this video, we explore the limit of (1-cos(x))/x as x approaches 0 and show that it equals 0. We use the Pythagorean trigonometric identity, algebraic manipulation, and the known limit of sin(x)/x as x approaches 0 to prove this result. This concept is helpful for understanding the derivative of sin(x). video https://cdn.kastatic.org/ka_thumbnails_cache/d16610b8-ad7b-498b-9c47-a0fcf1567f8e_1280_720_base.png Limit of (1-cos(x))/x as x approaches 0 In this video, we explore the limit of (1-cos(x))/x as x approaches 0 and show that it equals 0. We use the Pythagorean trigonometric identity, algebraic manipulation, and the known limit of sin(x)/x as x approaches 0 to prove this result. This concept is helpful for understanding the derivative of sin(x). https://cdn.kastatic.org/ka-youtube-converted/Ojtwh8XDF8M.mp4/Ojtwh8XDF8M.mp4 245 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-2/v/calculus-derivatives-1-new-hd-version 2023-07-27T01:47:53.920337803Z Formal definition of the derivative as a limit Discover how to define the derivative of a function at a specific point using the limit of the slope of the secant line. We'll explore the concept of finding the slope as the difference in function values approaches zero, represented by the limit of [f(c)-f(c+h)]/h as h→0. Sal Khan video https://cdn.kastatic.org/googleusercontent/f7edMdKGwytIJfVv9f8HZGykIr5fXzJ9eLpF7VFA2Y7DPQw7Zlq6wbdK1Rf2gsf5e8f8usxTd5xd5wz2nxJplqM Formal definition of the derivative as a limit Discover how to define the derivative of a function at a specific point using the limit of the slope of the secant line. We'll explore the concept of finding the slope as the difference in function values approaches zero, represented by the limit of [f(c)-f(c+h)]/h as h→0. https://cdn.kastatic.org/ka-youtube-converted/ANyVpMS3HL4.mp4/ANyVpMS3HL4.mp4 943 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-2/v/formal-and-alternate-form-of-the-derivative-for-ln-x 2023-07-27T01:47:53.920337803Z Worked example: Derivative as a limit Discover how to apply the formal and alternate forms of the derivative in real-world scenarios. We'll explore the process of finding the slope of tangent lines using both methods and compare their effectiveness in solving calculus problems. Let's dive into the practical side of derivatives to deepen our understanding. Sal Khan video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Worked example: Derivative as a limit Discover how to apply the formal and alternate forms of the derivative in real-world scenarios. We'll explore the process of finding the slope of tangent lines using both methods and compare their effectiveness in solving calculus problems. Let's dive into the practical side of derivatives to deepen our understanding. https://cdn.kastatic.org/ka-youtube-converted/m8yC7kR5Fuk.mp4/m8yC7kR5Fuk.mp4 346 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-2/v/calculus-derivatives-2-new-hd-version 2023-07-27T01:47:53.920337803Z The derivative of x² at x=3 using the formal definition Discover how to find the derivative of x² at x=3 using the formal definition of a derivative. Learn to calculate the slope of the tangent line at a specific point on the curve y=x² by applying the limit as the change in x approaches zero. This method helps determine the instantaneous rate of change for the function. Sal Khan video https://cdn.kastatic.org/googleusercontent/7GrhdQkwrs7A27TUTajFr6g7ara32pga7Zkj0WA6RfN1HOpeLOyspfnmtfsOn6sO8Yy_dEPUs-aS0lj6Jmm1Fu3P The derivative of x² at x=3 using the formal definition Discover how to find the derivative of x² at x=3 using the formal definition of a derivative. Learn to calculate the slope of the tangent line at a specific point on the curve y=x² by applying the limit as the change in x approaches zero. This method helps determine the instantaneous rate of change for the function. https://cdn.kastatic.org/ka-youtube-converted/IePCHjMeFkE.mp4/IePCHjMeFkE.mp4 508 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-5/v/power-rule 2023-07-27T01:47:53.920337803Z Power rule Let's dive into the power rule, a handy tool for finding the derivative of xⁿ. This rule simplifies the process of taking derivatives, especially for polynomials, by bringing the exponent out front and decrementing the power. We explore examples with positive, negative, and fractional exponents. Sal Khan video https://cdn.kastatic.org/googleusercontent/ep8pwcOTRuIUd4vjC92ZocRoYYB4y_bepmbnrR4FiJ25ltdCz6L4Az5ca1qhdj7L7rKY0yAazbXDABwd40BLLL48 Power rule Let's dive into the power rule, a handy tool for finding the derivative of xⁿ. This rule simplifies the process of taking derivatives, especially for polynomials, by bringing the exponent out front and decrementing the power. We explore examples with positive, negative, and fractional exponents. https://cdn.kastatic.org/ka-youtube-converted/bRZmfc1YFsQ.mp4/bRZmfc1YFsQ.mp4 234 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-6a/v/derivative-properties-and-polynomial-derivatives 2023-07-27T01:47:53.920337803Z Basic derivative rules Let's explore how to find the derivative of any polynomial using the power rule and additional properties. The derivative of a constant is always 0, and we can pull out a scalar constant when taking the derivative. Furthermore, the derivative of a sum of two functions is simply the sum of their derivatives. Sal Khan video https://cdn.kastatic.org/googleusercontent/518kcwioa0AcVYYwRMuZ53FfjtCKnXqyp7_jKbrCaJimmfPcewLLBSTVJyOM5tJnAEqj1CFkQpPjMUHJjkhb3Jc Basic derivative rules Let's explore how to find the derivative of any polynomial using the power rule and additional properties. The derivative of a constant is always 0, and we can pull out a scalar constant when taking the derivative. Furthermore, the derivative of a sum of two functions is simply the sum of their derivatives. https://cdn.kastatic.org/ka-youtube-converted/mzOBlH32qdk.mp4/mzOBlH32qdk.mp4 593 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-8/v/applying-the-product-rule-for-derivatives 2023-07-27T01:47:53.920337803Z Product rule Discover the product rule, a fundamental technique for finding the derivative of a function expressed as a product of two functions. We'll learn how to apply this rule to simplify differentiation and enhance our understanding of calculus. Sal Khan video https://cdn.kastatic.org/googleusercontent/gvxut2GAowO-jcx3gFpqOrHgtqwIaXYASRloUizkAbyX3oyOqCb455SbDI8VD4HmrHtgv0_qW1PsPqg3Pb0es1xV Product rule Discover the product rule, a fundamental technique for finding the derivative of a function expressed as a product of two functions. We'll learn how to apply this rule to simplify differentiation and enhance our understanding of calculus. https://cdn.kastatic.org/ka-youtube-converted/79ngr0Bur38.mp4/79ngr0Bur38.mp4 160 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-8/v/differentiating-products 2023-07-27T01:47:53.920337803Z Differentiating products We explore the product rule by finding the derivative of eˣcos(x). We identify eˣ as the first function and cos(x) as the second, then apply the product rule to calculate the derivative, simplifying our result for a clearer understanding. video https://cdn.kastatic.org/ka_thumbnails_cache/44c67ae6-dddf-4707-a4dd-af68bf6613b1_1280_720_base.png Differentiating products We explore the product rule by finding the derivative of eˣcos(x). We identify eˣ as the first function and cos(x) as the second, then apply the product rule to calculate the derivative, simplifying our result for a clearer understanding. https://cdn.kastatic.org/ka-youtube-converted/WxTrxxW0qeM.mp4/WxTrxxW0qeM.mp4 268 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-9/v/quotient-rule 2023-07-27T01:47:53.920337803Z Quotient rule Discover the quotient rule, a powerful technique for finding the derivative of a function expressed as a quotient. We'll explore how to apply this rule by differentiating the numerator and denominator functions, and then combining them to simplify the result. video https://cdn.kastatic.org/ka_thumbnails_cache/c14f1e21-de50-4dc6-baec-a95ea5c0be20_1280_720_base.png Quotient rule Discover the quotient rule, a powerful technique for finding the derivative of a function expressed as a quotient. We'll explore how to apply this rule by differentiating the numerator and denominator functions, and then combining them to simplify the result. https://cdn.kastatic.org/ka-youtube-converted/WqzY3xibFL8.mp4/WqzY3xibFL8.mp4 233 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-6b/v/differentiating-polynomials-example 2023-07-27T01:47:53.920337803Z Differentiating polynomials Let's explore how to differentiate polynomials using the power rule and derivative properties. We work with the function f(x)=x⁵+2x³-x² and apply the power rule to find its derivative, f'(x)=5x⁴+6x²-2x. Next, we evaluate f'(x) at x=2, determining that f'(2)=100, which represents the rate of change or slope of the tangent line at the specified point. video https://cdn.kastatic.org/ka_thumbnails_cache/0733d876-f0c2-49a1-a06a-4553cc763424_1280_720_base.png Differentiating polynomials Let's explore how to differentiate polynomials using the power rule and derivative properties. We work with the function f(x)=x⁵+2x³-x² and apply the power rule to find its derivative, f'(x)=5x⁴+6x²-2x. Next, we evaluate f'(x) at x=2, determining that f'(2)=100, which represents the rate of change or slope of the tangent line at the specified point. https://cdn.kastatic.org/ka-youtube-converted/-CTaxKTzbEI.mp4/-CTaxKTzbEI.mp4 399 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-7/v/sine-and-cosine-differentiation 2023-07-27T01:47:53.920337803Z Worked example: Derivatives of sin(x) and cos(x) Dive into the derivative of the function g(x) = 7sin(x) - 3cos(x) - (π/∛x)². By applying the power rule and the derivatives of sine and cosine functions, we efficiently determine the derivative g'(x) = 7cos(x) + 3sin(x) + 2π²/3 * x^(-5/3). Through algebraic manipulation and careful attention to detail, we tackle the problem's initially intimidating appearance. video https://cdn.kastatic.org/ka_thumbnails_cache/8ceec945-79e9-4fb2-87fc-be0e3e17ff4a_1280_720_base.png Worked example: Derivatives of sin(x) and cos(x) Dive into the derivative of the function g(x) = 7sin(x) - 3cos(x) - (π/∛x)². By applying the power rule and the derivatives of sine and cosine functions, we efficiently determine the derivative g'(x) = 7cos(x) + 3sin(x) + 2π²/3 * x^(-5/3). Through algebraic manipulation and careful attention to detail, we tackle the problem's initially intimidating appearance. https://cdn.kastatic.org/ka-youtube-converted/Iur13MNO0Ro.mp4/Iur13MNO0Ro.mp4 314 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-10/v/derivatives-of-tanx-and-cotx 2023-07-27T01:47:53.920337803Z Derivatives of tan(x) and cot(x) We find the derivatives of tan(x) and cot(x) by rewriting them as quotients of sin(x) and cos(x). Using the quotient rule, we determine that the derivative of tan(x) is sec^2(x) and the derivative of cot(x) is -csc^2(x). This process involves applying the Pythagorean identity to simplify final results. video https://cdn.kastatic.org/ka_thumbnails_cache/398fc34d-7f9d-477b-a994-1d426b066ef9_1280_720_base.png Derivatives of tan(x) and cot(x) We find the derivatives of tan(x) and cot(x) by rewriting them as quotients of sin(x) and cos(x). Using the quotient rule, we determine that the derivative of tan(x) is sec^2(x) and the derivative of cot(x) is -csc^2(x). This process involves applying the Pythagorean identity to simplify final results. https://cdn.kastatic.org/ka-youtube-converted/Rr_1GQyiRYs.mp4/Rr_1GQyiRYs.mp4 278 Limits and Derivatives https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-10/v/derivatives-of-secx-and-cscx 2023-07-27T01:47:53.920337803Z Derivatives of sec(x) and csc(x) Let's explore the derivatives of sec(x) and csc(x) by expressing them as 1/cos(x) and 1/sin(x), respectively, and applying the quotient rule. We discover that the derivative of sec(x) can be written as sin(x)/cos²(x) or tan(x)sec(x), and the derivative of csc(x) can be expressed as -cos(x)/sin²(x) or -cot(x)csc(x). video https://cdn.kastatic.org/ka_thumbnails_cache/71aa4205-5f9b-457e-b9f4-29e95f3e932b_1280_720_base.png Derivatives of sec(x) and csc(x) Let's explore the derivatives of sec(x) and csc(x) by expressing them as 1/cos(x) and 1/sin(x), respectively, and applying the quotient rule. We discover that the derivative of sec(x) can be written as sin(x)/cos²(x) or tan(x)sec(x), and the derivative of csc(x) can be expressed as -cos(x)/sin²(x) or -cot(x)csc(x). https://cdn.kastatic.org/ka-youtube-converted/TDJ5nXWEkWM.mp4/TDJ5nXWEkWM.mp4 268 Limits and Derivatives https://www.khanacademy.org/math/ka-revision-class-11-math/x64e5efbb29d87707:week-3-revision-ncert-class-11-math/x64e5efbb29d87707:limits-and-derivatives-revision-ncert-class-11-math/e/revision-limits-and-derivatives-ncert-class-11-math 2025-01-22T10:49:48.897158697Z Revision Limits and Derivatives Practice problems for revision of Limits and Derivatives. Satya Swaroop exercise