Calculus Examples

$f (x) = \sin (x^{2})$

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

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

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 $n = 2$ .

$\cos (x^{2}) (2 x)$

Step 1.2.2

Reorder the factors of $\cos (x^{2}) (2 x)$ .

$f' (x) = 2 x \cos (x^{2})$

Step 2

Find the second derivative.

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Step 2.1

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

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

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

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

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Differentiate using the Power Rule.

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Step 2.4.1

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

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

Step 2.4.2

Multiply $2$ by $- 1$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Add $1$ and $1$ .

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Simplify.

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Step 2.11.1

Apply the distributive property.

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

Step 2.11.2

Multiply $- 2$ by $2$ .

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

Step 3

Find the third derivative.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

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

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

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

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Step 3.2.3.1

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

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

Step 3.2.3.2

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

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

Step 3.2.3.3

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

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

Step 3.2.4

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

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

Step 3.2.5

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

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

Step 3.2.6

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

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Step 3.2.6.1

Move $x$ .

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

Step 3.2.6.2

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

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Step 3.2.6.2.1

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

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

Step 3.2.6.2.2

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

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

Step 3.2.6.3

Add $1$ and $2$ .

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

Step 3.2.7

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

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

Step 3.3

Evaluate $\frac{d}{d x} [2 \cos (x^{2})]$ .

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Step 3.3.1

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

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

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

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Step 3.3.2.1

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

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

Step 3.3.2.2

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

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

Step 3.3.2.3

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

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

Step 3.3.3

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

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

Step 3.3.4

Multiply $2$ by $- 1$ .

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

Step 3.3.5

Multiply $- 2$ by $2$ .

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

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

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

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Multiply $2$ by $- 4$ .

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

Step 3.4.2.2

Multiply $2$ by $- 4$ .

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

Step 3.4.2.3

Subtract $4 \sin (x^{2}) x$ from $- 8 \sin (x^{2}) x$ .

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

Step 3.4.3

Reorder terms.

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

Step 4

The third derivative of $f (x)$ with respect to $x$ is $- 8 x^{3} \cos (x^{2}) - 12 x \sin (x^{2})$ .

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