Calculus Examples

$\log_{7} (x) + 3 x^{5} - 4^{x}$

Step 1

By the Sum Rule, the derivative of $\log_{7} (x) + 3 x^{5} - 4^{x}$ with respect to $x$ is $\frac{d}{d x} [\log_{7} (x)] + \frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [- 4^{x}]$ .

$\frac{d}{d x} [\log_{7} (x)] + \frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [- 4^{x}]$

Step 2

The derivative of $\log_{7} (x)$ with respect to $x$ is $\frac{1}{x \ln (7)}$ .

$\frac{1}{x \ln (7)} + \frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [- 4^{x}]$

Step 3

Evaluate $\frac{d}{d x} [3 x^{5}]$ .

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Step 3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{5}$ with respect to $x$ is $3 \frac{d}{d x} [x^{5}]$ .

$\frac{1}{x \ln (7)} + 3 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 4^{x}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{1}{x \ln (7)} + 3 (5 x^{4}) + \frac{d}{d x} [- 4^{x}]$

Step 3.3

Multiply $5$ by $3$ .

$\frac{1}{x \ln (7)} + 15 x^{4} + \frac{d}{d x} [- 4^{x}]$

Step 4

Evaluate $\frac{d}{d x} [- 4^{x}]$ .

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Step 4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 4^{x}$ with respect to $x$ is $- \frac{d}{d x} [4^{x}]$ .

$\frac{1}{x \ln (7)} + 15 x^{4} - \frac{d}{d x} [4^{x}]$

Step 4.2

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

$\frac{1}{x \ln (7)} + 15 x^{4} - 4^{x} \ln (4)$

Step 5

Reorder terms.

$15 x^{4} - 4^{x} \ln (4) + \frac{1}{x \ln (7)}$