Calculus Examples

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

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 $g (x) = \cos (x)$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $\cos (x)$ .

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

Step 1.1.2

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

$f' (x) = 2 u \frac{d}{d x} (\cos (x))$

Step 1.1.3

Replace all occurrences of $u$ with $\cos (x)$ .

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

Step 1.2

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

$f' (x) = 2 \cos (x) (- \sin (x))$

Step 1.3

Multiply $- 1$ by $2$ .

$f' (x) = - 2 \cos (x) \sin (x)$

Step 2

Find the second derivative.

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Step 2.1

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

$f'' (x) = - 2 \frac{d}{d x} (\cos (x) \sin (x))$

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 $f (x) = \cos (x)$ and $g (x) = \sin (x)$ .

$f'' (x) = - 2 (\cos (x) \frac{d}{d x} (\sin (x)) + \sin (x) \frac{d}{d x} (\cos (x)))$

Step 2.3

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

$f'' (x) = - 2 (\cos (x) \cos (x) + \sin (x) \frac{d}{d x} (\cos (x)))$

Step 2.4

Raise $\cos (x)$ to the power of $1$ .

$f'' (x) = - 2 (\cos (x) \cos (x) + \sin (x) \frac{d}{d x} (\cos (x)))$

Step 2.5

Raise $\cos (x)$ to the power of $1$ .

$f'' (x) = - 2 (\cos (x) \cos (x) + \sin (x) \frac{d}{d x} (\cos (x)))$

Step 2.6

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

$f'' (x) = - 2 ({\cos (x)}^{1 + 1} + \sin (x) \frac{d}{d x} (\cos (x)))$

Step 2.7

Add $1$ and $1$ .

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

Step 2.8

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

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

Step 2.9

Raise $\sin (x)$ to the power of $1$ .

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

Step 2.10

Raise $\sin (x)$ to the power of $1$ .

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

Step 2.11

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

$f'' (x) = - 2 (\cos^{2} (x) - {\sin (x)}^{1 + 1})$

Step 2.12

Add $1$ and $1$ .

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

Step 2.13

Simplify.

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Step 2.13.1

Apply the distributive property.

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

Step 2.13.2

Multiply $- 1$ by $- 2$ .

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

Step 3

Find the third derivative.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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Step 3.2.2.1

To apply the Chain Rule, set $u_{1}$ as $\cos (x)$ .

$f''' (x) = - 2 (\frac{d}{d u (1)} ({u_{1}}^{2}) \frac{d}{d x} (\cos (x))) + \frac{d}{d x} (2 \sin^{2} (x))$

Step 3.2.2.2

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

$f''' (x) = - 2 (2 u_{1} \frac{d}{d x} (\cos (x))) + \frac{d}{d x} (2 \sin^{2} (x))$

Step 3.2.2.3

Replace all occurrences of $u_{1}$ with $\cos (x)$ .

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

Step 3.2.3

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

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

Step 3.2.4

Multiply $- 1$ by $2$ .

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

Step 3.2.5

Multiply $- 2$ by $- 2$ .

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

Step 3.3

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

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Step 3.3.1

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

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

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

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Step 3.3.2.1

To apply the Chain Rule, set $u_{2}$ as $\sin (x)$ .

$f''' (x) = 4 \cos (x) \sin (x) + 2 (\frac{d}{d u (2)} ({u_{2}}^{2}) \frac{d}{d x} (\sin (x)))$

Step 3.3.2.2

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

$f''' (x) = 4 \cos (x) \sin (x) + 2 (2 u_{2} \frac{d}{d x} (\sin (x)))$

Step 3.3.2.3

Replace all occurrences of $u_{2}$ with $\sin (x)$ .

$f''' (x) = 4 \cos (x) \sin (x) + 2 (2 \sin (x) \frac{d}{d x} (\sin (x)))$

Step 3.3.3

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

$f''' (x) = 4 \cos (x) \sin (x) + 2 (2 \sin (x) \cos (x))$

Step 3.3.4

Multiply $2$ by $2$ .

$f''' (x) = 4 \cos (x) \sin (x) + 4 \sin (x) \cos (x)$

Step 3.4

Combine terms.

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Step 3.4.1

Reorder the factors of $4 \sin (x) \cos (x)$ .

$f''' (x) = 4 \cos (x) \sin (x) + 4 \cos (x) \sin (x)$

Step 3.4.2

Add $4 \cos (x) \sin (x)$ and $4 \cos (x) \sin (x)$ .

$f''' (x) = 8 \cos (x) \sin (x)$

Step 4

Find the fourth derivative.

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Step 4.1

Since $8$ is constant with respect to $x$ , the derivative of $8 \cos (x) \sin (x)$ with respect to $x$ is $8 \frac{d}{d x} [\cos (x) \sin (x)]$ .

$f^{4} (x) = 8 \frac{d}{d x} (\cos (x) \sin (x))$

Step 4.2

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 $f (x) = \cos (x)$ and $g (x) = \sin (x)$ .

$f^{4} (x) = 8 (\cos (x) \frac{d}{d x} (\sin (x)) + \sin (x) \frac{d}{d x} (\cos (x)))$

Step 4.3

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

$f^{4} (x) = 8 (\cos (x) \cos (x) + \sin (x) \frac{d}{d x} (\cos (x)))$

Step 4.4

Raise $\cos (x)$ to the power of $1$ .

$f^{4} (x) = 8 (\cos (x) \cos (x) + \sin (x) \frac{d}{d x} (\cos (x)))$

Step 4.5

Raise $\cos (x)$ to the power of $1$ .

$f^{4} (x) = 8 (\cos (x) \cos (x) + \sin (x) \frac{d}{d x} (\cos (x)))$

Step 4.6

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

$f^{4} (x) = 8 ({\cos (x)}^{1 + 1} + \sin (x) \frac{d}{d x} (\cos (x)))$

Step 4.7

Add $1$ and $1$ .

$f^{4} (x) = 8 (\cos^{2} (x) + \sin (x) \frac{d}{d x} (\cos (x)))$

Step 4.8

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

$f^{4} (x) = 8 (\cos^{2} (x) + \sin (x) (- \sin (x)))$

Step 4.9

Raise $\sin (x)$ to the power of $1$ .

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

Step 4.10

Raise $\sin (x)$ to the power of $1$ .

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

Step 4.11

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

$f^{4} (x) = 8 (\cos^{2} (x) - {\sin (x)}^{1 + 1})$

Step 4.12

Add $1$ and $1$ .

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

Step 4.13

Simplify.

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Step 4.13.1

Apply the distributive property.

$f^{4} (x) = 8 \cos^{2} (x) + 8 (- \sin^{2} (x))$

Step 4.13.2

Multiply $- 1$ by $8$ .

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

Step 5

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

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