Calculus Examples

$6 \ln (x) - 2^{x}$

Step 1

By the Sum Rule, the derivative of $6 \ln (x) - 2^{x}$ with respect to $x$ is $\frac{d}{d x} [6 \ln (x)] + \frac{d}{d x} [- 2^{x}]$ .

$\frac{d}{d x} [6 \ln (x)] + \frac{d}{d x} [- 2^{x}]$

Step 2

Evaluate $\frac{d}{d x} [6 \ln (x)]$ .

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Step 2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 \ln (x)$ with respect to $x$ is $6 \frac{d}{d x} [\ln (x)]$ .

$6 \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [- 2^{x}]$

Step 2.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$6 \frac{1}{x} + \frac{d}{d x} [- 2^{x}]$

Step 2.3

Combine $6$ and $\frac{1}{x}$ .

$\frac{6}{x} + \frac{d}{d x} [- 2^{x}]$

Step 3

Evaluate $\frac{d}{d x} [- 2^{x}]$ .

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 2^{x}$ with respect to $x$ is $- \frac{d}{d x} [2^{x}]$ .

$\frac{6}{x} - \frac{d}{d x} [2^{x}]$

Step 3.2

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

$\frac{6}{x} - 2^{x} \ln (2)$

Step 4

Reorder terms.

$- 2^{x} \ln (2) + \frac{6}{x}$