Calculus Examples

$f (x) = - 6 x (4 x^{2}) - 1$

Step 1

By the Sum Rule, the derivative of $- 6 x (4 x^{2}) - 1$ with respect to $x$ is $\frac{d}{d x} [- 6 x (4 x^{2})] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [- 6 x (4 x^{2})] + \frac{d}{d x} [- 1]$

Step 2

Evaluate $\frac{d}{d x} [- 6 x (4 x^{2})]$ .

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Step 2.1

Multiply $4$ by $- 6$ .

$\frac{d}{d x} [- 24 x \cdot x^{2}] + \frac{d}{d x} [- 1]$

Step 2.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 2.2.1

Move $x^{2}$ .

$\frac{d}{d x} [- 24 (x^{2} x)] + \frac{d}{d x} [- 1]$

Step 2.2.2

Multiply $x^{2}$ by $x$ .

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Step 2.2.2.1

Raise $x$ to the power of $1$ .

$\frac{d}{d x} [- 24 (x^{2} x^{1})] + \frac{d}{d x} [- 1]$

Step 2.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d x} [- 24 x^{2 + 1}] + \frac{d}{d x} [- 1]$

Step 2.2.3

Add $2$ and $1$ .

$\frac{d}{d x} [- 24 x^{3}] + \frac{d}{d x} [- 1]$

Step 2.3

Since $- 24$ is constant with respect to $x$ , the derivative of $- 24 x^{3}$ with respect to $x$ is $- 24 \frac{d}{d x} [x^{3}]$ .

$- 24 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 1]$

Step 2.4

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

$- 24 (3 x^{2}) + \frac{d}{d x} [- 1]$

Step 2.5

Multiply $3$ by $- 24$ .

$- 72 x^{2} + \frac{d}{d x} [- 1]$

Step 3

Differentiate using the Constant Rule.

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$- 72 x^{2} + 0$

Step 3.2

Add $- 72 x^{2}$ and $0$ .

$- 72 x^{2}$