Calculus Examples

$\frac{d}{d x} \cdot (3 x^{10} - 5 x^{4} - \frac{2}{7})$

Step 1

This derivative could not be completed using the product rule. Mathway will use another method.

Step 2

By the Sum Rule, the derivative of $3 x^{10} - 5 x^{4} - \frac{2}{7}$ with respect to $x$ is $\frac{d}{d x} [3 x^{10}] + \frac{d}{d x} [- 5 x^{4}] + \frac{d}{d x} [- \frac{2}{7}]$ .

$\frac{d}{d x} [3 x^{10}] + \frac{d}{d x} [- 5 x^{4}] + \frac{d}{d x} [- \frac{2}{7}]$

Step 3

Evaluate $\frac{d}{d x} [3 x^{10}]$ .

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Step 3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{10}$ with respect to $x$ is $3 \frac{d}{d x} [x^{10}]$ .

$3 \frac{d}{d x} [x^{10}] + \frac{d}{d x} [- 5 x^{4}] + \frac{d}{d x} [- \frac{2}{7}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 10$ .

$3 (10 x^{9}) + \frac{d}{d x} [- 5 x^{4}] + \frac{d}{d x} [- \frac{2}{7}]$

Step 3.3

Multiply $10$ by $3$ .

$30 x^{9} + \frac{d}{d x} [- 5 x^{4}] + \frac{d}{d x} [- \frac{2}{7}]$

Step 4

Evaluate $\frac{d}{d x} [- 5 x^{4}]$ .

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Step 4.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{4}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{4}]$ .

$30 x^{9} - 5 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{2}{7}]$

Step 4.2

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

$30 x^{9} - 5 (4 x^{3}) + \frac{d}{d x} [- \frac{2}{7}]$

Step 4.3

Multiply $4$ by $- 5$ .

$30 x^{9} - 20 x^{3} + \frac{d}{d x} [- \frac{2}{7}]$

Step 5

Differentiate using the Constant Rule.

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Step 5.1

Since $- \frac{2}{7}$ is constant with respect to $x$ , the derivative of $- \frac{2}{7}$ with respect to $x$ is $0$ .

$30 x^{9} - 20 x^{3} + 0$

Step 5.2

Add $30 x^{9} - 20 x^{3}$ and $0$ .

$30 x^{9} - 20 x^{3}$