Calculus Examples

$g (x) = (x^{4} - 16) (x^{2} - 4)$

Step 1

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 $f (x) = x^{4} - 16$ and $g (x) = x^{2} - 4$ .

$(x^{4} - 16) \frac{d}{d x} [x^{2} - 4] + (x^{2} - 4) \frac{d}{d x} [x^{4} - 16]$

Step 2

Differentiate.

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Step 2.1

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

$(x^{4} - 16) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]) + (x^{2} - 4) \frac{d}{d x} [x^{4} - 16]$

Step 2.2

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

$(x^{4} - 16) (2 x + \frac{d}{d x} [- 4]) + (x^{2} - 4) \frac{d}{d x} [x^{4} - 16]$

Step 2.3

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

$(x^{4} - 16) (2 x + 0) + (x^{2} - 4) \frac{d}{d x} [x^{4} - 16]$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $2 x$ and $0$ .

$(x^{4} - 16) (2 x) + (x^{2} - 4) \frac{d}{d x} [x^{4} - 16]$

Step 2.4.2

Move $2$ to the left of $x^{4} - 16$ .

$2 \cdot (x^{4} - 16) x + (x^{2} - 4) \frac{d}{d x} [x^{4} - 16]$

Step 2.5

By the Sum Rule, the derivative of $x^{4} - 16$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 16]$ .

$2 (x^{4} - 16) x + (x^{2} - 4) (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 16])$

Step 2.6

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

$2 (x^{4} - 16) x + (x^{2} - 4) (4 x^{3} + \frac{d}{d x} [- 16])$

Step 2.7

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16$ with respect to $x$ is $0$ .

$2 (x^{4} - 16) x + (x^{2} - 4) (4 x^{3} + 0)$

Step 2.8

Simplify the expression.

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Step 2.8.1

Add $4 x^{3}$ and $0$ .

$2 (x^{4} - 16) x + (x^{2} - 4) (4 x^{3})$

Step 2.8.2

Move $4$ to the left of $x^{2} - 4$ .

$2 (x^{4} - 16) x + 4 \cdot (x^{2} - 4) x^{3}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$(2 x^{4} + 2 \cdot - 16) x + 4 (x^{2} - 4) x^{3}$

Step 3.2

Apply the distributive property.

$2 x^{4} x + 2 \cdot - 16 x + 4 (x^{2} - 4) x^{3}$

Step 3.3

Apply the distributive property.

$2 x^{4} x + 2 \cdot - 16 x + (4 x^{2} + 4 \cdot - 4) x^{3}$

Step 3.4

Apply the distributive property.

$2 x^{4} x + 2 \cdot - 16 x + 4 x^{2} x^{3} + 4 \cdot - 4 x^{3}$

Step 3.5

Combine terms.

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Step 3.5.1

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

$2 (x^{1} x^{4}) + 2 \cdot - 16 x + 4 x^{2} x^{3} + 4 \cdot - 4 x^{3}$

Step 3.5.2

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

$2 x^{1 + 4} + 2 \cdot - 16 x + 4 x^{2} x^{3} + 4 \cdot - 4 x^{3}$

Step 3.5.3

Add $1$ and $4$ .

$2 x^{5} + 2 \cdot - 16 x + 4 x^{2} x^{3} + 4 \cdot - 4 x^{3}$

Step 3.5.4

Multiply $2$ by $- 16$ .

$2 x^{5} - 32 x + 4 x^{2} x^{3} + 4 \cdot - 4 x^{3}$

Step 3.5.5

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 3.5.5.1

Move $x^{3}$ .

$2 x^{5} - 32 x + 4 (x^{3} x^{2}) + 4 \cdot - 4 x^{3}$

Step 3.5.5.2

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

$2 x^{5} - 32 x + 4 x^{3 + 2} + 4 \cdot - 4 x^{3}$

Step 3.5.5.3

Add $3$ and $2$ .

$2 x^{5} - 32 x + 4 x^{5} + 4 \cdot - 4 x^{3}$

Step 3.5.6

Multiply $4$ by $- 4$ .

$2 x^{5} - 32 x + 4 x^{5} - 16 x^{3}$

Step 3.5.7

Add $2 x^{5}$ and $4 x^{5}$ .

$6 x^{5} - 32 x - 16 x^{3}$

Step 3.6

Reorder terms.

$6 x^{5} - 16 x^{3} - 32 x$