Calculus Examples

$f (x) = 3 x^{5} - 5 x^{3} + 4$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $5$ by $3$ .

$15 x^{4} + \frac{d}{d x} [- 5 x^{3}] + \frac{d}{d x} [4]$

Step 3

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

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Step 3.1

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

$15 x^{4} - 5 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [4]$

Step 3.2

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

$15 x^{4} - 5 (3 x^{2}) + \frac{d}{d x} [4]$

Step 3.3

Multiply $3$ by $- 5$ .

$15 x^{4} - 15 x^{2} + \frac{d}{d x} [4]$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$15 x^{4} - 15 x^{2} + 0$

Step 4.2

Add $15 x^{4} - 15 x^{2}$ and $0$ .

$15 x^{4} - 15 x^{2}$