Calculus Examples

$3 {(4)}^{x} + 5 e^{x}$

Step 1

By the Sum Rule, the derivative of $3 {(4)}^{x} + 5 e^{x}$ with respect to $x$ is $\frac{d}{d x} [3 {(4)}^{x}] + \frac{d}{d x} [5 e^{x}]$ .

$\frac{d}{d x} [3 {(4)}^{x}] + \frac{d}{d x} [5 e^{x}]$

Step 2

Evaluate $\frac{d}{d x} [3 {(4)}^{x}]$ .

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Step 2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 \cdot 4^{x}$ with respect to $x$ is $3 \frac{d}{d x} [4^{x}]$ .

$3 \frac{d}{d x} [4^{x}] + \frac{d}{d x} [5 e^{x}]$

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $4$ .

$3 \cdot 4^{x} \ln (4) + \frac{d}{d x} [5 e^{x}]$

Step 3

Evaluate $\frac{d}{d x} [5 e^{x}]$ .

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Step 3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 e^{x}$ with respect to $x$ is $5 \frac{d}{d x} [e^{x}]$ .

$3 \cdot 4^{x} \ln (4) + 5 \frac{d}{d x} [e^{x}]$

Step 3.2

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

$3 \cdot 4^{x} \ln (4) + 5 e^{x}$