Calculus Examples

$2^{x} - 3^{x}$

Step 1

By the Sum Rule, the derivative of $2^{x} - 3^{x}$ with respect to $x$ is $\frac{d}{d x} [2^{x}] + \frac{d}{d x} [- 3^{x}]$ .

$\frac{d}{d x} [2^{x}] + \frac{d}{d x} [- 3^{x}]$

Step 2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $2$ .

$2^{x} \ln (2) + \frac{d}{d x} [- 3^{x}]$

Step 3

Evaluate $\frac{d}{d x} [- 3^{x}]$ .

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 3^{x}$ with respect to $x$ is $- \frac{d}{d x} [3^{x}]$ .

$2^{x} \ln (2) - \frac{d}{d x} [3^{x}]$

Step 3.2

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

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