Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} - 4$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $x^{2} - 4$ .

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

Step 1.1.2

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

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

Step 1.1.3

Replace all occurrences of $u$ with $x^{2} - 4$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $2 x$ and $0$ .

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

Step 1.2.4.2

Multiply $2$ by $2$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Combine terms.

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Step 1.3.3.1

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

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

Step 1.3.3.2

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

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

Step 1.3.3.3

Add $1$ and $2$ .

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

Step 1.3.3.4

Multiply $4$ by $- 4$ .

$f' (x) = 4 x^{3} - 16 x$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $4 x^{3} - 16 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 16 x]$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [- 16 x]$

Step 2.3

Evaluate $\frac{d}{d x} [- 16 x]$ .

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Step 2.3.1

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16 x$ with respect to $x$ is $- 16 \frac{d}{d x} [x]$ .

$12 x^{2} - 16 \frac{d}{d x} [x]$

Step 2.3.2

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

$12 x^{2} - 16 \cdot 1$

Step 2.3.3

Multiply $- 16$ by $1$ .

$f'' (x) = 12 x^{2} - 16$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $12 x^{2} - 16$ with respect to $x$ is $\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 16]$ .

$\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 16]$

Step 3.2

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

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Step 3.2.1

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

$12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 16]$

Step 3.2.2

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

$12 (2 x) + \frac{d}{d x} [- 16]$

Step 3.2.3

Multiply $2$ by $12$ .

$24 x + \frac{d}{d x} [- 16]$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

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

$24 x + 0$

Step 3.3.2

Add $24 x$ and $0$ .

$f''' (x) = 24 x$

Step 4

Find the fourth derivative.

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Step 4.1

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

$24 \frac{d}{d x} [x]$

Step 4.2

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

$24 \cdot 1$

Step 4.3

Multiply $24$ by $1$ .

$f^{4} (x) = 24$