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Second derivative of $$$5^{x}$$$
Related calculators: Derivative Calculator, Logarithmic Differentiation Calculator
Your Input
Find $$$\frac{d^{2}}{dx^{2}} \left(5^{x}\right)$$$.
Solution
Find the first derivative $$$\frac{d}{dx} \left(5^{x}\right)$$$
Apply the exponential rule $$$\frac{d}{dx} \left(n^{x}\right) = n^{x} \ln\left(n\right)$$$ with $$$n = 5$$$:
$${\color{red}\left(\frac{d}{dx} \left(5^{x}\right)\right)} = {\color{red}\left(5^{x} \ln\left(5\right)\right)}$$Thus, $$$\frac{d}{dx} \left(5^{x}\right) = 5^{x} \ln\left(5\right)$$$.
Next, $$$\frac{d^{2}}{dx^{2}} \left(5^{x}\right) = \frac{d}{dx} \left(5^{x} \ln\left(5\right)\right)$$$
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 $$$c = \ln\left(5\right)$$$ and $$$f{\left(x \right)} = 5^{x}$$$:
$${\color{red}\left(\frac{d}{dx} \left(5^{x} \ln\left(5\right)\right)\right)} = {\color{red}\left(\ln\left(5\right) \frac{d}{dx} \left(5^{x}\right)\right)}$$Apply the exponential rule $$$\frac{d}{dx} \left(n^{x}\right) = n^{x} \ln\left(n\right)$$$ with $$$n = 5$$$:
$$\ln\left(5\right) {\color{red}\left(\frac{d}{dx} \left(5^{x}\right)\right)} = \ln\left(5\right) {\color{red}\left(5^{x} \ln\left(5\right)\right)}$$Thus, $$$\frac{d}{dx} \left(5^{x} \ln\left(5\right)\right) = 5^{x} \ln^{2}\left(5\right)$$$.
Therefore, $$$\frac{d^{2}}{dx^{2}} \left(5^{x}\right) = 5^{x} \ln^{2}\left(5\right)$$$.
Answer
$$$\frac{d^{2}}{dx^{2}} \left(5^{x}\right) = 5^{x} \ln^{2}\left(5\right)$$$A