https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:continuity-and-differentiability-ncert-new/x6f66cbee9a2f805b:untitled-887/v/where-a-function-is-not-differentiable 2020-06-23T07:22:41Z Differentiability at a point (old) An older video where Sal finds the points on the graph of a function where the function isn't differentiable. Sal Khan video https://cdn.kastatic.org/googleusercontent/XHAd6gYSQhfNoXFEYuv8KY1BI5OYxPAKMeb380wNeZ4BulBKolYJxVcfhAzSWqDTY6ofpneKXgjIlAswV93SAMsj Differentiability at a point (old) An older video where Sal finds the points on the graph of a function where the function isn't differentiable. https://cdn.kastatic.org/ka-youtube-converted/pj1VvscsV0s.mp4/pj1VvscsV0s.mp4 215 Differentiability https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-4/v/differentiability-at-a-point-algebraic-not-differentiable 2023-07-27T02:34:53.22139958Z Differentiability at a point: algebraic (function isn't differentiable) We examine a piecewise function to determine its differentiability and continuity at an edge point. Through analyzing left and right-hand limits, we find that the function is continuous at the point. However, due to differing slopes from the left and right, the function is not differentiable at the edge point. video https://cdn.kastatic.org/ka_thumbnails_cache/28a81671-858e-4bf2-adf6-3dc89b196a7e_1280_720_base.png Differentiability at a point: algebraic (function isn't differentiable) We examine a piecewise function to determine its differentiability and continuity at an edge point. Through analyzing left and right-hand limits, we find that the function is continuous at the point. However, due to differing slopes from the left and right, the function is not differentiable at the edge point. https://cdn.kastatic.org/ka-youtube-converted/BpPHve1cCCs.mp4/BpPHve1cCCs.mp4 382 Differentiability https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-4/e/differentiability-at-a-point-algebraic 2025-05-05T08:14:25.350149109Z Differentiability at a point: algebraic Analyze algebraic functions to determine whether they are continuous and/or differentiable at a given point. Tomer Gal exercise https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-4/a/proof-differentiability-implies-continuity 2021-04-29T07:57:16Z Proof: Differentiability implies continuity article 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 Differentiability https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab-3-1a/v/chain-rule-introduction 2021-03-30T19:55:55Z Chain rule The chain rule states that the derivative of f(g(x)) is f'(g(x))⋅g'(x). In other words, it helps us differentiate *composite functions*. For example, sin(x²) is a composite function because it can be constructed as f(g(x)) for f(x)=sin(x) and g(x)=x². Using the chain rule and the derivatives of sin(x) and x², we can then find the derivative of sin(x²). Sal Khan video https://cdn.kastatic.org/googleusercontent/u4Y4Eqtrlf1A6dKUR2uXvjSZGo5CWBzjC639ioWX3Ta-ZGKkzgWD1F_SEcr9Y98mtlBx89TTHiPCZXzqT-I_ZMY Chain rule The chain rule states that the derivative of f(g(x)) is f'(g(x))⋅g'(x). In other words, it helps us differentiate *composite functions*. For example, sin(x²) is a composite function because it can be constructed as f(g(x)) for f(x)=sin(x) and g(x)=x². Using the chain rule and the derivatives of sin(x) and x², we can then find the derivative of sin(x²). https://cdn.kastatic.org/ka-youtube-converted/0T0QrHO56qg.mp4/0T0QrHO56qg.mp4 307 Differentiability https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab-3-1a/e/differentiate-composite-functions-intro 2021-04-29T07:57:28Z Chain rule intro Use the chain rule to differentiate composite functions like sin(2x+1) or [cos(x)]³. Tomer Gal exercise https://www.khanacademy.org/math/grade-12-math-snc-aligned/x35fdd21198d6755b:higher-order-derivatives/x35fdd21198d6755b:chain-rule/a/chain-rule-overview 2021-04-29T07:54:28Z Chain rule overview A quick overview sheet of the chain rule. article https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:continuity-and-differentiability-ncert-new/x6f66cbee9a2f805b:untitled-887/e/differentiate-composite-functions 2025-03-12T14:20:10.072734823Z Differentiate composite functions (all function types) Find derivatives of functions that are the composition of two (more basic) functions using the chain rule. Tomer Gal exercise https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab-3-5b/v/applying-the-chain-rule-and-product-rule 2024-08-20T23:45:10.987222765Z Applying the chain rule and product rule Let's explore multiple strategies to tackle derivatives involving both the product and chain rules. We start by applying the chain rule first, then the product rule. Next, we switch it up, using the product rule first, then the chain rule. Discover how these different approaches lead us to the same answer, demonstrating the flexibility and interconnectedness of these mathematical rules. video https://cdn.kastatic.org/ka_thumbnails_cache/5944fb24-452d-4084-9dd1-d1bbdabebb96_1280_720_base.png Applying the chain rule and product rule Let's explore multiple strategies to tackle derivatives involving both the product and chain rules. We start by applying the chain rule first, then the product rule. Next, we switch it up, using the product rule first, then the chain rule. Discover how these different approaches lead us to the same answer, demonstrating the flexibility and interconnectedness of these mathematical rules. https://cdn.kastatic.org/ka-youtube-converted/QqAPCIIdx-E.mp4/QqAPCIIdx-E.mp4 354 Differentiability https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab-3-1b/e/exp-log-trig-radical-composite-diff 2025-05-05T08:14:25.350149109Z Chain rule capstone Find the derivatives of exponential, logarithmic, trigonometric, and radical functions, whose inner function is a polynomial. Tomer Gal exercise https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab-3-2/v/implicit-differentiation-1 2020-07-26T09:23:45Z Implicit differentiation Some relationships cannot be represented by an explicit function. For example, x²+y²=1. Implicit differentiation helps us find ​dy/dx even for relationships like that. This is done using the chain ​rule, and viewing y as an implicit function of x. For example, according to the chain rule, the derivative of y² would be 2y⋅(dy/dx). Sal Khan video https://cdn.kastatic.org/googleusercontent/J3IuW_SdB3YQgggI2OZHsoN8vmEIRuzq-U6_vX3OPTsL_FymniZfpKb5rUJhtAjMv_E0A_kP_FUEUZ8QVQmS1hHJ Implicit differentiation Some relationships cannot be represented by an explicit function. For example, x²+y²=1. Implicit differentiation helps us find ​dy/dx even for relationships like that. This is done using the chain ​rule, and viewing y as an implicit function of x. For example, according to the chain rule, the derivative of y² would be 2y⋅(dy/dx). https://cdn.kastatic.org/ka-youtube-converted/mSVrqKZDRF4.mp4/mSVrqKZDRF4.mp4 482 Differentiability https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab-3-2/e/implicit-differentiation 2025-05-05T08:14:25.350149109Z Implicit differentiation Practice using implicit differentiation. Tomer Gal exercise https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab-3-4/v/derivative-inverse-sine 2023-07-27T02:34:53.22139958Z Derivative of inverse sine Unlock the mystery of the derivative of inverse sine! Let's dive into the world of calculus, rearranging equations and applying implicit differentiation to find the derivative of y with respect to x. Using trigonometric identities, we transform the derivative into a function of x, revealing a fascinating relationship. Sal Khan video https://cdn.kastatic.org/googleusercontent/TC-D-ES3wUzt8Qplru972T5gPY59R5SLsMvo-hWbmKjf9Rnhlt9RcNuPt23kV_0NB5hrYnpqTXsd-O4wP32xctFOig Derivative of inverse sine Unlock the mystery of the derivative of inverse sine! Let's dive into the world of calculus, rearranging equations and applying implicit differentiation to find the derivative of y with respect to x. Using trigonometric identities, we transform the derivative into a function of x, revealing a fascinating relationship. https://cdn.kastatic.org/ka-youtube-converted/yIQUhXa-n-M.mp4/yIQUhXa-n-M.mp4 296 Differentiability https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab-3-4/v/derivative-inverse-cosine 2023-07-27T02:34:53.22139958Z Derivative of inverse cosine Unraveling the mystery of the inverse cosine function, we find its derivative equals -1/(sqrt(1 - x^2)). This step-by-step proof guides us to a fascinating comparison with the derivative of inverse sine, revealing a captivating connection between these two trigonometric functions. Sal Khan video https://cdn.kastatic.org/googleusercontent/xD9q4NJ8qH9IMBxQhEvVVZrvt43R-vuoPevtrTigfTAGX82bI1XBDAEmCzu26a_B63_ZCRAqSQn2sAEvXWhwi58 Derivative of inverse cosine Unraveling the mystery of the inverse cosine function, we find its derivative equals -1/(sqrt(1 - x^2)). This step-by-step proof guides us to a fascinating comparison with the derivative of inverse sine, revealing a captivating connection between these two trigonometric functions. https://cdn.kastatic.org/ka-youtube-converted/v_OfFmMRvOc.mp4/v_OfFmMRvOc.mp4 224 Differentiability https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab-3-4/v/derivative-inverse-tangent 2023-07-27T02:34:53.22139958Z Derivative of inverse tangent Now let's explore the derivative of the inverse tangent function. Starting with the derivative of tangent, we use the chain rule and trigonometric identities to find the derivative of its inverse. Join us as we investigate this fascinating mathematical process! Sal Khan video https://cdn.kastatic.org/googleusercontent/f0bVGN6g1oYXxST0OHEmXjlgAhdT_fYwKolK1lXe4QrIJ4-QMJ4cCGtHmTvRYRdbJQ87mFs0vYt77vKcsK9a0F6OeQ Derivative of inverse tangent Now let's explore the derivative of the inverse tangent function. Starting with the derivative of tangent, we use the chain rule and trigonometric identities to find the derivative of its inverse. Join us as we investigate this fascinating mathematical process! https://cdn.kastatic.org/ka-youtube-converted/G7WyEp8gHs0.mp4/G7WyEp8gHs0.mp4 362 Differentiability https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab-3-4/e/derivatives-of-inverse-trigonometric-functions 2025-05-05T08:14:25.350149109Z Derivatives of inverse trigonometric functions Differentiate functions that contain the inverse trigonometric functions arcsin(x), arccos(x), and arctan(x). Tomer Gal exercise