# 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.

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

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

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