- [[derivative - logarithmic functions]] # Idea Take the exponential function: $ f(x)=a^{x} $ Its derivative is the function itself, $a^x$ multiplied by the derivative of the power, $x$, and natural log of the base, $ln(a)$: $f^{\prime}(x)=a^{x} x^\prime \ln (a) $ Note the special case: $ f(x)=e^{x} $ $f^{\prime}(x)=e^{x} x^\prime \ln (e)$ $f^{\prime}(x)=e^{x} x^0 \ln(e)$ $f^{\prime}(x)=e^{x}$ because $ln(e) = 1$. ## Example 1 $ y=4^{3 x^{2}} $ $y^\prime = 4^{3x^2} (3x^2)^\prime \ln(4)$ $y^\prime = 4^{3x^2} 6x \ln(4)$ ## Example 2 $y = 2^x e^x$ $y\prime = 2^x ln(2)e^x + 2^xe^x ln(e)$ $y\prime = 2^x ln(2)e^x + 2^xe^x$ $y\prime = 2^x e^x (ln(2) + 1)$ Apply [[product rule]] here. # References - [Derivatives of Exponential Functions - YouTube](https://www.youtube.com/watch?v=yg_497u6JnA) - [Derivatives of Exponential Functions](https://math24.net/derivatives-exponential-functions.html) - [Differentiation of Exponential Functions](https://www.analyzemath.com/calculus/Differentiation/exponential.html) - [Derivative of Exponential Function - Formula, Proof, Examples](https://www.cuemath.com/calculus/derivative-of-exponential-function/)