Step By Step Calculus » 10.3 - Derivatives of Exponential and Logarithmic Functions

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Synopsis
Using the definition of derivative, we can find a formula for the derivative of any exponential function. Then we can apply the inverse function rule to find a formula for the derivative of any logarithmic function.
  • \dfrac{d}{dx}a^x=a^x\cdot \ln a\dfrac{d}{dx}a^x=a^x\cdot \ln a.
  • \dfrac{d}{dx}\log_a(x)=\dfrac{1}{x\ln a}\dfrac{d}{dx}\log_a(x)=\dfrac{1}{x\ln a}.
Combining with the Derivative Chain rule, we have
Derivative Exponential Rule:\dfrac{d}{dx}a^{f(x)}=a^{f(x)}f'(x) \ln a\dfrac{d}{dx}a^{f(x)}=a^{f(x)}f'(x) \ln a
Derivative Logarithmic Rule:\dfrac{d}{dx}\log_a(f(x))=\dfrac{f'(x)}{f(x) \ln a}\dfrac{d}{dx}\log_a(f(x))=\dfrac{f'(x)}{f(x) \ln a}
These rules allow us to develop the following:
Derivative General Power Rule:\dfrac{d}{dx}(f(x))^p=p(f(x))^{p-1}f'(x)\dfrac{d}{dx}(f(x))^p=p(f(x))^{p-1}f'(x) for f(x) \ge 0f(x) \ge 0 and p \in \mathbb{R}p \in \mathbb{R}
We can use the Derivative Chain rule and the Derivative Exponential rule to write a general formula for the derivative of f(x)^{g(x)}f(x)^{g(x)} when f(x) \ge 0f(x) \ge 0.
Derivative PowEx Rule:\dfrac{d}{dx}f(x)^{g(x)}=g(x)f(x)^{g(x)-1}f^\prime(x)+\ln(f(x))f(x)^{g(x)}g^\prime(x)\dfrac{d}{dx}f(x)^{g(x)}=g(x)f(x)^{g(x)-1}f^\prime(x)+\ln(f(x))f(x)^{g(x)}g^\prime(x).
This new list of rules can be used to find the derivative of even more complicated funcitons.