Calculus Examples

$4 x (x^{2} - 3)$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $4 x (x^{2} - 3)$ with respect to $x$ is $4 \frac{d}{d x} [x (x^{2} - 3)]$ .

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

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x^{2} - 3$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $2 x$ and $0$ .

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

Step 4

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

$4 (2 (x^{1} x) + (x^{2} - 3) \frac{d}{d x} [x])$

Step 5

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

$4 (2 (x^{1} x^{1}) + (x^{2} - 3) \frac{d}{d x} [x])$

Step 6

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

$4 (2 x^{1 + 1} + (x^{2} - 3) \frac{d}{d x} [x])$

Step 7

Add $1$ and $1$ .

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

Step 8

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

$4 (2 x^{2} + (x^{2} - 3) \cdot 1)$

Step 9

Simplify by adding terms.

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Step 9.1

Multiply $x^{2} - 3$ by $1$ .

$4 (2 x^{2} + x^{2} - 3)$

Step 9.2

Add $2 x^{2}$ and $x^{2}$ .

$4 (3 x^{2} - 3)$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$4 (3 x^{2}) + 4 \cdot - 3$

Step 10.2

Combine terms.

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Step 10.2.1

Multiply $3$ by $4$ .

$12 x^{2} + 4 \cdot - 3$

Step 10.2.2

Multiply $4$ by $- 3$ .

$12 x^{2} - 12$