https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab-3-1b/v/exponential-functions-differentiation-intro 2023-07-27T02:34:53.22139958Z Derivative of aˣ (for any positive base a) Let's dive deeper into the fascinating world of derivatives, specifically focusing on the derivative of aˣ for any positive base a. Using the derivative of eˣ and the chain rule, we unravel the mystery behind differentiating exponential functions. We then apply our newfound knowledge to differentiate the expression 8⋅3ˣ.   video https://cdn.kastatic.org/ka_thumbnails_cache/1f5ebcc7-0f50-400d-b7d2-8809eb454d8d_1280_720_base.png Derivative of aˣ (for any positive base a) Let's dive deeper into the fascinating world of derivatives, specifically focusing on the derivative of aˣ for any positive base a. Using the derivative of eˣ and the chain rule, we unravel the mystery behind differentiating exponential functions. We then apply our newfound knowledge to differentiate the expression 8⋅3ˣ.   https://cdn.kastatic.org/ka-youtube-converted/gHzLHknEk1M.mp4/gHzLHknEk1M.mp4 325 Logarithmic differentiation https://www.khanacademy.org/math/differential-calculus/dc-chain/dc-logarithmic-diff/v/calculus-derivative-of-x-x-x 2021-04-13T17:49:50Z Composite exponential function differentiation Sal finds the derivatives of xˣ and x^(xˣ). They are surprisingly fun to do! Sal Khan video https://cdn.kastatic.org/googleusercontent/0MpzPWEFfcYegqyHgzckEEX6dsbhs67DvnMKkF6MDeTmZ99uF4jAe14qc6F2tFEgg4hDzyaqIW03Er0OI-Z9eRnM Composite exponential function differentiation Sal finds the derivatives of xˣ and x^(xˣ). They are surprisingly fun to do! https://cdn.kastatic.org/ka-youtube-converted/N5kkwVoAtkc.mp4/N5kkwVoAtkc.mp4 542 Logarithmic differentiation https://www.khanacademy.org/math/differential-calculus/dc-chain/dc-logarithmic-diff/e/logarithmic-differentiation 2025-05-05T08:14:25.350149109Z Composite exponential function differentiation Differentiate functions that have the variable both in the base and in the exponent. For example, differentiate xˣ. Tomer Gal exercise https://www.khanacademy.org/math/revision-term-1-ncert-math-class-12/x6a634f0b79812b7f:week-2/x6a634f0b79812b7f:continuity-and-differentiability/v/log-differentiation-of-implicit-functions 2025-05-22T13:12:03.767544849Z Log differentiation of implicit functions In this video, we learn how to perform logarithmic differentiation of implicit functions. We first look at the function x^y = y^x and then pick up (cosx)^y = (cosx)^y. In both of these, the powers can be brought down using properties of log. This is what makes logarithmeic differentiation a useful too for situations like these. Once we've dealt with the powers, we can use product rule and chain rule to perform the differentiation. Once this part is done, we can readily isolate dy/dx. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Log differentiation of implicit functions In this video, we learn how to perform logarithmic differentiation of implicit functions. We first look at the function x^y = y^x and then pick up (cosx)^y = (cosx)^y. In both of these, the powers can be brought down using properties of log. This is what makes logarithmeic differentiation a useful too for situations like these. Once we've dealt with the powers, we can use product rule and chain rule to perform the differentiation. Once this part is done, we can readily isolate dy/dx. https://cdn.kastatic.org/ka-youtube-converted/uF08_lFLEOQ.mp4/uF08_lFLEOQ.mp4 476 Logarithmic differentiation https://www.khanacademy.org/math/revision-term-1-ncert-math-class-12/x6a634f0b79812b7f:week-2/x6a634f0b79812b7f:continuity-and-differentiability/e/logarithmic-differentiation-of-implicit-functions 2025-08-08T02:45:51.78671331Z Logarithmic differentiation of implicit functions Logarithmic differentiation of implicit functions Ashish Gupta exercise