Second derivative of $3^{x}$

The calculator will find the second derivative of $3^{x}$, with steps shown.

Related calculators: Derivative Calculator, Logarithmic Differentiation Calculator

Leave empty for autodetection.
Leave empty, if you don't need the derivative at a specific point.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Find $\frac{d^{2}}{dx^{2}} \left(3^{x}\right)$.

Find the first derivative $\frac{d}{dx} \left(3^{x}\right)$

Apply the exponential rule $\frac{d}{dx} \left(n^{x}\right) = n^{x} \ln\left(n\right)$ with $n = 3$:

$${\color{red}\left(\frac{d}{dx} \left(3^{x}\right)\right)} = {\color{red}\left(3^{x} \ln\left(3\right)\right)}$$

Thus, $\frac{d}{dx} \left(3^{x}\right) = 3^{x} \ln\left(3\right)$.

Next, $\frac{d^{2}}{dx^{2}} \left(3^{x}\right) = \frac{d}{dx} \left(3^{x} \ln\left(3\right)\right)$

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 $c = \ln\left(3\right)$ and $f{\left(x \right)} = 3^{x}$:

$${\color{red}\left(\frac{d}{dx} \left(3^{x} \ln\left(3\right)\right)\right)} = {\color{red}\left(\ln\left(3\right) \frac{d}{dx} \left(3^{x}\right)\right)}$$

Apply the exponential rule $\frac{d}{dx} \left(n^{x}\right) = n^{x} \ln\left(n\right)$ with $n = 3$:

$$\ln\left(3\right) {\color{red}\left(\frac{d}{dx} \left(3^{x}\right)\right)} = \ln\left(3\right) {\color{red}\left(3^{x} \ln\left(3\right)\right)}$$

Thus, $\frac{d}{dx} \left(3^{x} \ln\left(3\right)\right) = 3^{x} \ln^{2}\left(3\right)$.

Therefore, $\frac{d^{2}}{dx^{2}} \left(3^{x}\right) = 3^{x} \ln^{2}\left(3\right)$.

$\frac{d^{2}}{dx^{2}} \left(3^{x}\right) = 3^{x} \ln^{2}\left(3\right)$A