Calculus Examples

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

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

The second derivative of $f (x)$ with respect to $x$ is $- 4 x^{2} \cos (x^{2}) - 2 \sin (x^{2})$ .

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