Math Sc E Derivative Of Chain Rule (Video), Simple Examples Of Using The Chain Rule

I need help with exponential functions. I know that the derivative of $e^x$ is $e^x$, but wolfram alpha shows a different answer to my function below. If you, for example, take the derivative of $e^{-2x}$ do you get $-2e^{-2x}$ or $e^{-2x}$?



You have to apply the chain rule: if $f(x)$ is a differentiable function then the derivative of $e^{f(x)}$ is $f”(x)e^{f(x)}$.

Đang xem: E derivative of chain rule


You have to use the chain rule here. Writing $f(x) = e^x$ and $g(x) = -2x$ we have $h(x) := f(g(x)) = e^{-2x}$, hence by the chain rule$$ h”(x) = f”(g(x))g”(x) $$Now $f”(x) = e^x$, hence $f”(g(x)) = e^{-2x}$, and $g”(x) = -2$, this gives$$ h”(x) = f”(g(x))g”(x) = e^{-2x} cdot (-2) = -2e^{-2x} $$


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Differentiating powers of e where the power is an exponential function like $e^{a^x}$ when $a$ is a constant


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