Calculus Examples

$y = \cos (X^{2})$

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 $f (X) = \cos (X)$ and $g (X) = X^{2}$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $X^{2}$ .

$\frac{d}{d u} [\cos (u)] \frac{d}{d X} [X^{2}]$

Step 1.1.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

$- \sin (u) \frac{d}{d X} [X^{2}]$

Step 1.1.3

Replace all occurrences of $u$ with $X^{2}$ .

$- \sin (X^{2}) \frac{d}{d X} [X^{2}]$

Step 1.2

Differentiate using the Power Rule.

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Step 1.2.1

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

$- \sin (X^{2}) (2 X)$

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

Multiply $2$ by $- 1$ .

$- 2 \sin (X^{2}) X$

Step 1.2.2.2

Reorder the factors of $- 2 \sin (X^{2}) X$ .

$f' (X) = - 2 X \sin (X^{2})$

Step 2

Find the second derivative.

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Step 2.1

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

$- 2 \frac{d}{d X} [X \sin (X^{2})]$

Step 2.2

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 $g (X) = \sin (X^{2})$ .

$- 2 (X \frac{d}{d X} [\sin (X^{2})] + \sin (X^{2}) \frac{d}{d X} [X])$

Step 2.3

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

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Step 2.3.1

To apply the Chain Rule, set $u$ as $X^{2}$ .

$- 2 (X (\frac{d}{d u} [\sin (u)] \frac{d}{d X} [X^{2}]) + \sin (X^{2}) \frac{d}{d X} [X])$

Step 2.3.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$- 2 (X (\cos (u) \frac{d}{d X} [X^{2}]) + \sin (X^{2}) \frac{d}{d X} [X])$

Step 2.3.3

Replace all occurrences of $u$ with $X^{2}$ .

$- 2 (X (\cos (X^{2}) \frac{d}{d X} [X^{2}]) + \sin (X^{2}) \frac{d}{d X} [X])$

Step 2.4

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

$- 2 (X (\cos (X^{2}) (2 X)) + \sin (X^{2}) \frac{d}{d X} [X])$

Step 2.5

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

$- 2 (X^{1} X (\cos (X^{2}) \cdot (2)) + \sin (X^{2}) \frac{d}{d X} [X])$

Step 2.6

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

$- 2 (X^{1} X^{1} (\cos (X^{2}) \cdot (2)) + \sin (X^{2}) \frac{d}{d X} [X])$

Step 2.7

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

$- 2 (X^{1 + 1} (\cos (X^{2}) \cdot (2)) + \sin (X^{2}) \frac{d}{d X} [X])$

Step 2.8

Simplify the expression.

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Step 2.8.1

Add $1$ and $1$ .

$- 2 (X^{2} (\cos (X^{2}) \cdot (2)) + \sin (X^{2}) \frac{d}{d X} [X])$

Step 2.8.2

Move $2$ to the left of $\cos (X^{2})$ .

$- 2 (X^{2} (2 \cdot \cos (X^{2})) + \sin (X^{2}) \frac{d}{d X} [X])$

Step 2.9

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

$- 2 (X^{2} (2 \cos (X^{2})) + \sin (X^{2}) \cdot 1)$

Step 2.10

Multiply $\sin (X^{2})$ by $1$ .

$- 2 (X^{2} (2 \cos (X^{2})) + \sin (X^{2}))$

Step 2.11

Simplify.

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Step 2.11.1

Apply the distributive property.

$- 2 (X^{2} (2 \cos (X^{2}))) - 2 \sin (X^{2})$

Step 2.11.2

Multiply $2$ by $- 2$ .

$f'' (X) = - 4 X^{2} \cos (X^{2}) - 2 \sin (X^{2})$

Step 3

Find the third derivative.

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Step 3.1

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

$\frac{d}{d X} [- 4 X^{2} \cos (X^{2})] + \frac{d}{d X} [- 2 \sin (X^{2})]$

Step 3.2

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

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Step 3.2.1

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

$- 4 \frac{d}{d X} [X^{2} \cos (X^{2})] + \frac{d}{d X} [- 2 \sin (X^{2})]$

Step 3.2.2

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 $f (X) = X^{2}$ and $g (X) = \cos (X^{2})$ .

$- 4 (X^{2} \frac{d}{d X} [\cos (X^{2})] + \cos (X^{2}) \frac{d}{d X} [X^{2}]) + \frac{d}{d X} [- 2 \sin (X^{2})]$

Step 3.2.3

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

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Step 3.2.3.1

To apply the Chain Rule, set $u_{1}$ as $X^{2}$ .

$- 4 (X^{2} (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d X} [X^{2}]) + \cos (X^{2}) \frac{d}{d X} [X^{2}]) + \frac{d}{d X} [- 2 \sin (X^{2})]$

Step 3.2.3.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$- 4 (X^{2} (- \sin (u_{1}) \frac{d}{d X} [X^{2}]) + \cos (X^{2}) \frac{d}{d X} [X^{2}]) + \frac{d}{d X} [- 2 \sin (X^{2})]$

Step 3.2.3.3

Replace all occurrences of $u_{1}$ with $X^{2}$ .

$- 4 (X^{2} (- \sin (X^{2}) \frac{d}{d X} [X^{2}]) + \cos (X^{2}) \frac{d}{d X} [X^{2}]) + \frac{d}{d X} [- 2 \sin (X^{2})]$

Step 3.2.4

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

$- 4 (X^{2} (- \sin (X^{2}) (2 X)) + \cos (X^{2}) \frac{d}{d X} [X^{2}]) + \frac{d}{d X} [- 2 \sin (X^{2})]$

Step 3.2.5

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

$- 4 (X^{2} (- \sin (X^{2}) (2 X)) + \cos (X^{2}) (2 X)) + \frac{d}{d X} [- 2 \sin (X^{2})]$

Step 3.2.6

Multiply $2$ by $- 1$ .

$- 4 (X^{2} (- 2 \sin (X^{2}) X) + \cos (X^{2}) (2 X)) + \frac{d}{d X} [- 2 \sin (X^{2})]$

Step 3.2.7

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

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Step 3.2.7.1

Move $X$ .

$- 4 (X \cdot X^{2} (- 2 \sin (X^{2})) + \cos (X^{2}) (2 X)) + \frac{d}{d X} [- 2 \sin (X^{2})]$

Step 3.2.7.2

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

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Step 3.2.7.2.1

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

$- 4 (X^{1} X^{2} (- 2 \sin (X^{2})) + \cos (X^{2}) (2 X)) + \frac{d}{d X} [- 2 \sin (X^{2})]$

Step 3.2.7.2.2

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

$- 4 (X^{1 + 2} (- 2 \sin (X^{2})) + \cos (X^{2}) (2 X)) + \frac{d}{d X} [- 2 \sin (X^{2})]$

Step 3.2.7.3

Add $1$ and $2$ .

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

Step 3.3

Evaluate $\frac{d}{d X} [- 2 \sin (X^{2})]$ .

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Step 3.3.1

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

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

Step 3.3.2

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

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Step 3.3.2.1

To apply the Chain Rule, set $u_{2}$ as $X^{2}$ .

$- 4 (X^{3} (- 2 \sin (X^{2})) + \cos (X^{2}) (2 X)) - 2 (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d X} [X^{2}])$

Step 3.3.2.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$- 4 (X^{3} (- 2 \sin (X^{2})) + \cos (X^{2}) (2 X)) - 2 (\cos (u_{2}) \frac{d}{d X} [X^{2}])$

Step 3.3.2.3

Replace all occurrences of $u_{2}$ with $X^{2}$ .

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

Step 3.3.3

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

$- 4 (X^{3} (- 2 \sin (X^{2})) + \cos (X^{2}) (2 X)) - 2 (\cos (X^{2}) (2 X))$

Step 3.3.4

Multiply $2$ by $- 2$ .

$- 4 (X^{3} (- 2 \sin (X^{2})) + \cos (X^{2}) (2 X)) - 4 \cos (X^{2}) X$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$- 4 (X^{3} (- 2 \sin (X^{2}))) - 4 (\cos (X^{2}) (2 X)) - 4 \cos (X^{2}) X$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Multiply $- 2$ by $- 4$ .

$8 (X^{3} (\sin (X^{2}))) - 4 (\cos (X^{2}) (2 X)) - 4 \cos (X^{2}) X$

Step 3.4.2.2

Multiply $2$ by $- 4$ .

$8 X^{3} \sin (X^{2}) - 8 (\cos (X^{2}) (X)) - 4 \cos (X^{2}) X$

Step 3.4.2.3

Subtract $4 \cos (X^{2}) X$ from $- 8 \cos (X^{2}) X$ .

$8 X^{3} \sin (X^{2}) - 12 \cos (X^{2}) X$

Step 3.4.3

Reorder terms.

$f''' (X) = 8 X^{3} \sin (X^{2}) - 12 X \cos (X^{2})$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $8 X^{3} \sin (X^{2}) - 12 X \cos (X^{2})$ with respect to $X$ is $\frac{d}{d X} [8 X^{3} \sin (X^{2})] + \frac{d}{d X} [- 12 X \cos (X^{2})]$ .

$\frac{d}{d X} [8 X^{3} \sin (X^{2})] + \frac{d}{d X} [- 12 X \cos (X^{2})]$

Step 4.2

Evaluate $\frac{d}{d X} [8 X^{3} \sin (X^{2})]$ .

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Step 4.2.1

Since $8$ is constant with respect to $X$ , the derivative of $8 X^{3} \sin (X^{2})$ with respect to $X$ is $8 \frac{d}{d X} [X^{3} \sin (X^{2})]$ .

$8 \frac{d}{d X} [X^{3} \sin (X^{2})] + \frac{d}{d X} [- 12 X \cos (X^{2})]$

Step 4.2.2

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 $f (X) = X^{3}$ and $g (X) = \sin (X^{2})$ .

$8 (X^{3} \frac{d}{d X} [\sin (X^{2})] + \sin (X^{2}) \frac{d}{d X} [X^{3}]) + \frac{d}{d X} [- 12 X \cos (X^{2})]$

Step 4.2.3

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

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Step 4.2.3.1

To apply the Chain Rule, set $u_{1}$ as $X^{2}$ .

$8 (X^{3} (\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d X} [X^{2}]) + \sin (X^{2}) \frac{d}{d X} [X^{3}]) + \frac{d}{d X} [- 12 X \cos (X^{2})]$

Step 4.2.3.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$8 (X^{3} (\cos (u_{1}) \frac{d}{d X} [X^{2}]) + \sin (X^{2}) \frac{d}{d X} [X^{3}]) + \frac{d}{d X} [- 12 X \cos (X^{2})]$

Step 4.2.3.3

Replace all occurrences of $u_{1}$ with $X^{2}$ .

$8 (X^{3} (\cos (X^{2}) \frac{d}{d X} [X^{2}]) + \sin (X^{2}) \frac{d}{d X} [X^{3}]) + \frac{d}{d X} [- 12 X \cos (X^{2})]$

Step 4.2.4

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

$8 (X^{3} (\cos (X^{2}) (2 X)) + \sin (X^{2}) \frac{d}{d X} [X^{3}]) + \frac{d}{d X} [- 12 X \cos (X^{2})]$

Step 4.2.5

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

$8 (X^{3} (\cos (X^{2}) (2 X)) + \sin (X^{2}) (3 X^{2})) + \frac{d}{d X} [- 12 X \cos (X^{2})]$

Step 4.2.6

Multiply $X^{3}$ by $X$ by adding the exponents.

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Step 4.2.6.1

Move $X$ .

$8 (X \cdot X^{3} (\cos (X^{2}) \cdot (2)) + \sin (X^{2}) (3 X^{2})) + \frac{d}{d X} [- 12 X \cos (X^{2})]$

Step 4.2.6.2

Multiply $X$ by $X^{3}$ .

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Step 4.2.6.2.1

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

$8 (X^{1} X^{3} (\cos (X^{2}) \cdot (2)) + \sin (X^{2}) (3 X^{2})) + \frac{d}{d X} [- 12 X \cos (X^{2})]$

Step 4.2.6.2.2

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

$8 (X^{1 + 3} (\cos (X^{2}) \cdot (2)) + \sin (X^{2}) (3 X^{2})) + \frac{d}{d X} [- 12 X \cos (X^{2})]$

Step 4.2.6.3

Add $1$ and $3$ .

$8 (X^{4} (\cos (X^{2}) \cdot (2)) + \sin (X^{2}) (3 X^{2})) + \frac{d}{d X} [- 12 X \cos (X^{2})]$

Step 4.2.7

Move $2$ to the left of $\cos (X^{2})$ .

$8 (X^{4} (2 \cos (X^{2})) + \sin (X^{2}) (3 X^{2})) + \frac{d}{d X} [- 12 X \cos (X^{2})]$

Step 4.3

Evaluate $\frac{d}{d X} [- 12 X \cos (X^{2})]$ .

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Step 4.3.1

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

$8 (X^{4} (2 \cos (X^{2})) + \sin (X^{2}) (3 X^{2})) - 12 \frac{d}{d X} [X \cos (X^{2})]$

Step 4.3.2

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 $g (X) = \cos (X^{2})$ .

$8 (X^{4} (2 \cos (X^{2})) + \sin (X^{2}) (3 X^{2})) - 12 (X \frac{d}{d X} [\cos (X^{2})] + \cos (X^{2}) \frac{d}{d X} [X])$

Step 4.3.3

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

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Step 4.3.3.1

To apply the Chain Rule, set $u_{2}$ as $X^{2}$ .

$8 (X^{4} (2 \cos (X^{2})) + \sin (X^{2}) (3 X^{2})) - 12 (X (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d X} [X^{2}]) + \cos (X^{2}) \frac{d}{d X} [X])$

Step 4.3.3.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$8 (X^{4} (2 \cos (X^{2})) + \sin (X^{2}) (3 X^{2})) - 12 (X (- \sin (u_{2}) \frac{d}{d X} [X^{2}]) + \cos (X^{2}) \frac{d}{d X} [X])$

Step 4.3.3.3

Replace all occurrences of $u_{2}$ with $X^{2}$ .

$8 (X^{4} (2 \cos (X^{2})) + \sin (X^{2}) (3 X^{2})) - 12 (X (- \sin (X^{2}) \frac{d}{d X} [X^{2}]) + \cos (X^{2}) \frac{d}{d X} [X])$

Step 4.3.4

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

$8 (X^{4} (2 \cos (X^{2})) + \sin (X^{2}) (3 X^{2})) - 12 (X (- \sin (X^{2}) (2 X)) + \cos (X^{2}) \frac{d}{d X} [X])$

Step 4.3.5

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

$8 (X^{4} (2 \cos (X^{2})) + \sin (X^{2}) (3 X^{2})) - 12 (X (- \sin (X^{2}) (2 X)) + \cos (X^{2}) \cdot 1)$

Step 4.3.6

Multiply $2$ by $- 1$ .

$8 (X^{4} (2 \cos (X^{2})) + \sin (X^{2}) (3 X^{2})) - 12 (X (- 2 \sin (X^{2}) X) + \cos (X^{2}) \cdot 1)$

Step 4.3.7

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

$8 (X^{4} (2 \cos (X^{2})) + \sin (X^{2}) (3 X^{2})) - 12 (X^{1} X (- 2 \sin (X^{2})) + \cos (X^{2}) \cdot 1)$

Step 4.3.8

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

$8 (X^{4} (2 \cos (X^{2})) + \sin (X^{2}) (3 X^{2})) - 12 (X^{1} X^{1} (- 2 \sin (X^{2})) + \cos (X^{2}) \cdot 1)$

Step 4.3.9

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

$8 (X^{4} (2 \cos (X^{2})) + \sin (X^{2}) (3 X^{2})) - 12 (X^{1 + 1} (- 2 \sin (X^{2})) + \cos (X^{2}) \cdot 1)$

Step 4.3.10

Add $1$ and $1$ .

$8 (X^{4} (2 \cos (X^{2})) + \sin (X^{2}) (3 X^{2})) - 12 (X^{2} (- 2 \sin (X^{2})) + \cos (X^{2}) \cdot 1)$

Step 4.3.11

Multiply $\cos (X^{2})$ by $1$ .

$8 (X^{4} (2 \cos (X^{2})) + \sin (X^{2}) (3 X^{2})) - 12 (X^{2} (- 2 \sin (X^{2})) + \cos (X^{2}))$

Step 4.4

Simplify.

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Step 4.4.1

Apply the distributive property.

$8 (X^{4} (2 \cos (X^{2}))) + 8 (\sin (X^{2}) (3 X^{2})) - 12 (X^{2} (- 2 \sin (X^{2})) + \cos (X^{2}))$

Step 4.4.2

Apply the distributive property.

$8 (X^{4} (2 \cos (X^{2}))) + 8 (\sin (X^{2}) (3 X^{2})) - 12 (X^{2} (- 2 \sin (X^{2}))) - 12 \cos (X^{2})$

Step 4.4.3

Combine terms.

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Step 4.4.3.1

Multiply $2$ by $8$ .

$16 (X^{4} (\cos (X^{2}))) + 8 (\sin (X^{2}) (3 X^{2})) - 12 (X^{2} (- 2 \sin (X^{2}))) - 12 \cos (X^{2})$

Step 4.4.3.2

Multiply $3$ by $8$ .

$16 X^{4} \cos (X^{2}) + 24 (\sin (X^{2}) (X^{2})) - 12 (X^{2} (- 2 \sin (X^{2}))) - 12 \cos (X^{2})$

Step 4.4.3.3

Multiply $- 2$ by $- 12$ .

$16 X^{4} \cos (X^{2}) + 24 \sin (X^{2}) X^{2} + 24 (X^{2} (\sin (X^{2}))) - 12 \cos (X^{2})$

Step 4.4.3.4

Add $24 \sin (X^{2}) X^{2}$ and $24 X^{2} \sin (X^{2})$ .

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Step 4.4.3.4.1

Move $\sin (X^{2})$ .

$16 X^{4} \cos (X^{2}) + 24 X^{2} \sin (X^{2}) + 24 X^{2} \sin (X^{2}) - 12 \cos (X^{2})$

Step 4.4.3.4.2

Add $24 X^{2} \sin (X^{2})$ and $24 X^{2} \sin (X^{2})$ .

$f^{4} (X) = 16 X^{4} \cos (X^{2}) + 48 X^{2} \sin (X^{2}) - 12 \cos (X^{2})$