Calculus Examples

$3 x^{4} - 4 x^{3} - 12 x^{2} + 5$

Step 1

By the Sum Rule, the derivative of $3 x^{4} - 4 x^{3} - 12 x^{2} + 5$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [5]$

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

$3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [5]$

Step 2.2

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

$3 (4 x^{3}) + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [5]$

Step 2.3

Multiply $4$ by $3$ .

$12 x^{3} + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [5]$

Step 3

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

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Step 3.1

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

$12 x^{3} - 4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [5]$

Step 3.2

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

$12 x^{3} - 4 (3 x^{2}) + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [5]$

Step 3.3

Multiply $3$ by $- 4$ .

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

Step 4

Evaluate $\frac{d}{d x} [- 12 x^{2}]$ .

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Step 4.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x^{2}$ with respect to $x$ is $- 12 \frac{d}{d x} [x^{2}]$ .

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

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 4.3

Multiply $2$ by $- 12$ .

$12 x^{3} - 12 x^{2} - 24 x + \frac{d}{d x} [5]$

Step 5

Differentiate using the Constant Rule.

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Step 5.1

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$12 x^{3} - 12 x^{2} - 24 x + 0$

Step 5.2

Add $12 x^{3} - 12 x^{2} - 24 x$ and $0$ .

$12 x^{3} - 12 x^{2} - 24 x$