Algebra Examples

$f (x) = 3^{x}$

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $3$ .

$f' (x) = 3^{x} \ln (3)$

Find the second derivative.

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Since $\ln (3)$ is constant with respect to $x$ , the derivative of $3^{x} \ln (3)$ with respect to $x$ is $\ln (3) \frac{d}{d x} [3^{x}]$ .

$f'' (x) = \ln (3) \frac{d}{d x} (3^{x})$

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $3$ .

$f'' (x) = \ln (3) (3^{x} \ln (3))$

Raise $\ln (3)$ to the power of $1$ .

$f'' (x) = 3^{x} ((\ln (3)) \ln (3))$

Raise $\ln (3)$ to the power of $1$ .

$f'' (x) = 3^{x} ((\ln (3)) (\ln (3)))$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = 3^{x} {(\ln (3))}^{1 + 1}$

Add $1$ and $1$ .

$f'' (x) = 3^{x} \ln^{2} (3)$

The second derivative of $f (x)$ with respect to $x$ is $3^{x} \ln^{2} (3)$ .

$3^{x} \ln^{2} (3)$