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Second derivative of $$$a^{x}$$$ with respect to $$$x$$$

The calculator will find the second derivative of $$$a^{x}$$$ with respect to $$$x$$$, with steps shown.

Related calculators: Derivative Calculator, Logarithmic Differentiation Calculator

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Your Input

Find $$$\frac{d^{2}}{dx^{2}} \left(a^{x}\right)$$$.

Solution

Find the first derivative $$$\frac{d}{dx} \left(a^{x}\right)$$$

Apply the exponential rule $$$\frac{d}{dx} \left(n^{x}\right) = n^{x} \ln\left(n\right)$$$ with $$$n = a$$$:

$${\color{red}\left(\frac{d}{dx} \left(a^{x}\right)\right)} = {\color{red}\left(a^{x} \ln\left(a\right)\right)}$$

Thus, $$$\frac{d}{dx} \left(a^{x}\right) = a^{x} \ln\left(a\right)$$$.

Next, $$$\frac{d^{2}}{dx^{2}} \left(a^{x}\right) = \frac{d}{dx} \left(a^{x} \ln\left(a\right)\right)$$$

Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = \ln\left(a\right)$$$ and $$$f{\left(x \right)} = a^{x}$$$:

$${\color{red}\left(\frac{d}{dx} \left(a^{x} \ln\left(a\right)\right)\right)} = {\color{red}\left(\ln\left(a\right) \frac{d}{dx} \left(a^{x}\right)\right)}$$

Apply the exponential rule $$$\frac{d}{dx} \left(n^{x}\right) = n^{x} \ln\left(n\right)$$$ with $$$n = a$$$:

$$\ln\left(a\right) {\color{red}\left(\frac{d}{dx} \left(a^{x}\right)\right)} = \ln\left(a\right) {\color{red}\left(a^{x} \ln\left(a\right)\right)}$$

Thus, $$$\frac{d}{dx} \left(a^{x} \ln\left(a\right)\right) = a^{x} \ln^{2}\left(a\right)$$$.

Therefore, $$$\frac{d^{2}}{dx^{2}} \left(a^{x}\right) = a^{x} \ln^{2}\left(a\right)$$$.

Answer

$$$\frac{d^{2}}{dx^{2}} \left(a^{x}\right) = a^{x} \ln^{2}\left(a\right)$$$A