Calculus Examples

$y = {5.4}^{x}$

Step 1

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $5.4$ .

$f' (x) = {5.4}^{x} \ln (5.4)$

Step 2

Find the second derivative.

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Step 2.1

Since $\ln (5.4)$ is constant with respect to $x$ , the derivative of ${5.4}^{x} \ln (5.4)$ with respect to $x$ is $\ln (5.4) \frac{d}{d x} [{5.4}^{x}]$ .

$\ln (5.4) \frac{d}{d x} [{5.4}^{x}]$

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $5.4$ .

$\ln (5.4) ({5.4}^{x} \ln (5.4))$

Step 2.3

Raise $\ln (5.4)$ to the power of $1$ .

${5.4}^{x} (\ln^{1} (5.4) \ln (5.4))$

Step 2.4

Raise $\ln (5.4)$ to the power of $1$ .

${5.4}^{x} (\ln^{1} (5.4) \ln^{1} (5.4))$

Step 2.5

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

${5.4}^{x} {\ln (5.4)}^{1 + 1}$

Step 2.6

Add $1$ and $1$ .

$f'' (x) = {5.4}^{x} \ln^{2} (5.4)$