Instantly find the derivative of exponential functions, including the base $e$ ($e^x$) and general base $b$ ($b^x$), with step-by-step solutions.
Applying Exponential and Chain Rules...
Exponential functions are critical in calculus, especially $f(x) = e^x$ and $f(x) = b^x$, which are used to describe continuous growth and decay processes. Our calculator provides instant, accurate derivatives for these functions, including cases requiring the Chain Rule.
The derivative process relies on two fundamental rules. Note that $\exp(x)$ is equivalent to $e^x$ in mathematical notation.
For composite functions, such as $f(x) = e^{g(x)}$ or $f(x) = b^{g(x)}$, the Chain Rule is mandatory. You differentiate the 'outside' function (the exponential) and then multiply by the derivative of the 'inside' function (the exponent $g(x)$).
$$\frac{d}{dx} (e^{g(x)}) = e^{g(x)} \cdot g'(x)$$ $$\frac{d}{dx} (b^{g(x)}) = b^{g(x)} \cdot \ln(b) \cdot g'(x)$$
For example, to find the derivative of $f(x) = e^{x^2}$: $$f'(x) = e^{x^2} \cdot \frac{d}{dx}(x^2) = e^{x^2} \cdot (2x) = 2x e^{x^2}$$
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