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 Derivatives Introduction https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-2/v/alternate-form-of-the-derivative 2023-07-27T01:47:53.920337803Z Formal and alternate form of the derivative Discover the two methods for defining the derivative of a function at a specific point using limit expressions. In this video we'll explore the standard form and alternate form, enhancing our understanding of tangent line slopes and secant line approximations. Sal Khan video https://cdn.kastatic.org/googleusercontent/RJ5ynAu__DEqYV6GmzDEfo0ZPvcyf5YkTi2W4gXktvh7WpA_ruD8LKnpPY1Ij_wxkB-2ym0HezKSnhR6QSshcbPX Formal and alternate form of the derivative Discover the two methods for defining the derivative of a function at a specific point using limit expressions. In this video we'll explore the standard form and alternate form, enhancing our understanding of tangent line slopes and secant line approximations. https://cdn.kastatic.org/ka-youtube-converted/Df2escG-Vu0.mp4/Df2escG-Vu0.mp4 293 Derivatives Introduction 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 Derivatives Introduction https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-2/v/formal-and-alternate-form-of-the-derivative-example-1 2023-07-27T01:47:53.920337803Z Worked example: Derivative from limit expression We explore a limit expression and discover that it represents the derivative of the function f(x) = x³ at the point x = 5. By analyzing the alternate form of the derivative, we gain a deeper understanding of tangent lines and their slopes. Sal Khan video https://cdn.kastatic.org/googleusercontent/mRb04hmxds73ZL_sMNXBuSvgJWIEdDWaFqII1RDcrdsyueWxuoNan8KXuqI5CEwLvxcoRGQOL2CXmaQbgNqDVYE Worked example: Derivative from limit expression We explore a limit expression and discover that it represents the derivative of the function f(x) = x³ at the point x = 5. By analyzing the alternate form of the derivative, we gain a deeper understanding of tangent lines and their slopes. https://cdn.kastatic.org/ka-youtube-converted/ePh8iCbcXfA.mp4/ePh8iCbcXfA.mp4 317 Derivatives Introduction 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 Derivatives Introduction https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:limits-and-derivatives-ncert-new/x0419e5b3b578592a:derivatives-introduction/v/derivatives-using-first-principle 2025-05-22T11:09:14.016569349Z Derivatives using first principle In this video, we'll solve two problems where we find the limit of functins from first principle. Through these problems, we'll learn the strategies that we can use to find limits in such problems. We first find the derivative of 1/x^2 and then of tan(x). In both of these, step 1 is to use the first principle and plug in the value for f(x). Step 2 is manipulation - algebraic or trigonometric. Step 3 is getting rid of h - either through direct plugging or by using some simpler functions limit. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Derivatives using first principle In this video, we'll solve two problems where we find the limit of functins from first principle. Through these problems, we'll learn the strategies that we can use to find limits in such problems. We first find the derivative of 1/x^2 and then of tan(x). In both of these, step 1 is to use the first principle and plug in the value for f(x). Step 2 is manipulation - algebraic or trigonometric. Step 3 is getting rid of h - either through direct plugging or by using some simpler functions limit. https://cdn.kastatic.org/ka-youtube-converted/HvrHNR26irM.mp4/HvrHNR26irM.mp4 329 Derivatives Introduction https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-2/e/the-formal-and-alternate-form-of-the-derivative 2025-03-01T15:52:37.13987578Z Derivative as a limit Write derivatives of functions as limit expressions. Tomer Gal exercise 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 Derivatives Introduction https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-5/e/power-rule-intro 2025-03-01T15:52:37.13987578Z Power rule (positive integer powers) Use the power rule to differentiate functions of the form xⁿ where n is a positive integer. Tomer Gal exercise https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-5/e/differentiate-negative-powers 2025-03-01T15:52:37.13987578Z Power rule (negative & fractional powers) Use the power rule to differentiate functions of the form xⁿ where n is a negative integer or a fraction. Tomer Gal exercise https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-5/v/power-rule-with-rewriting 2023-07-27T01:47:53.920337803Z Power rule (with rewriting the expression) Discover how the power rule helps us find derivatives of functions like 1/x, ∛x, or ∛x². By rewriting these functions as xⁿ, where n is a negative or fractional exponent, we can apply the power rule to calculate their derivatives with ease. video https://cdn.kastatic.org/googleusercontent/iloQtEvwi681cOiNQDTDAFZp0HEWF_UYwBJ5WnqTPL_QPzXbhoL8ubebuSSiWljErut5ASmmcQb9L0MdMu-_TCI Power rule (with rewriting the expression) Discover how the power rule helps us find derivatives of functions like 1/x, ∛x, or ∛x². By rewriting these functions as xⁿ, where n is a negative or fractional exponent, we can apply the power rule to calculate their derivatives with ease. https://cdn.kastatic.org/ka-youtube-converted/H-v4oraDjuM.mp4/H-v4oraDjuM.mp4 264 Derivatives Introduction https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-5/e/differentiate-fractional-powers 2025-03-01T15:52:37.13987578Z Power rule (with rewriting the expression) Find and evaluate the derivatives of expressions like ∜x³  and 1/x² by rewriting them in the form xⁿ and using the power rule. Tomer Gal exercise