Precalculus Examples

$f (x) = 3 x^{4} - 2 x^{6} + 23$

Step 1

By the Sum Rule, the derivative of $3 x^{4} - 2 x^{6} + 23$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 2 x^{6}] + \frac{d}{d x} [23]$ .

$\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 2 x^{6}] + \frac{d}{d x} [23]$

Step 2

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

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Step 2.1

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

$3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 2 x^{6}] + \frac{d}{d x} [23]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$3 (4 x^{3}) + \frac{d}{d x} [- 2 x^{6}] + \frac{d}{d x} [23]$

Step 2.3

Multiply $4$ by $3$ .

$12 x^{3} + \frac{d}{d x} [- 2 x^{6}] + \frac{d}{d x} [23]$

Step 3

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

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Step 3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{6}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{6}]$ .

$12 x^{3} - 2 \frac{d}{d x} [x^{6}] + \frac{d}{d x} [23]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$12 x^{3} - 2 (6 x^{5}) + \frac{d}{d x} [23]$

Step 3.3

Multiply $6$ by $- 2$ .

$12 x^{3} - 12 x^{5} + \frac{d}{d x} [23]$

Step 4

Since $23$ is constant with respect to $x$ , the derivative of $23$ with respect to $x$ is $0$ .

$12 x^{3} - 12 x^{5} + 0$

Step 5

Simplify.

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Step 5.1

Add $12 x^{3} - 12 x^{5}$ and $0$ .

$12 x^{3} - 12 x^{5}$

Step 5.2

Reorder terms.

$- 12 x^{5} + 12 x^{3}$