Calculus Examples

$y = 3 - 2 x^{3} + 4 x^{7}$

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [4 x^{7}]$

Step 2

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

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Step 2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{3}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{3}]$ .

$0 - 2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [4 x^{7}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$0 - 2 (3 x^{2}) + \frac{d}{d x} [4 x^{7}]$

Step 2.3

Multiply $3$ by $- 2$ .

$0 - 6 x^{2} + \frac{d}{d x} [4 x^{7}]$

Step 3

Evaluate $\frac{d}{d x} [4 x^{7}]$ .

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Step 3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{7}$ with respect to $x$ is $4 \frac{d}{d x} [x^{7}]$ .

$0 - 6 x^{2} + 4 \frac{d}{d x} [x^{7}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$0 - 6 x^{2} + 4 (7 x^{6})$

Step 3.3

Multiply $7$ by $4$ .

$0 - 6 x^{2} + 28 x^{6}$

Step 4

Simplify.

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Step 4.1

Subtract $6 x^{2}$ from $0$ .

$- 6 x^{2} + 28 x^{6}$

Step 4.2

Reorder terms.

$28 x^{6} - 6 x^{2}$