Calculus Examples

$f (x) = \sin (a x)$

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

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Step 1.1.1

To apply the Chain Rule, set $u$ as $a x$ .

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

Step 1.1.2

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

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

Step 1.1.3

Replace all occurrences of $u$ with $a x$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

$\cos (a x) (a \cdot 1)$

Step 1.2.3

Simplify the expression.

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Step 1.2.3.1

Multiply $a$ by $1$ .

$\cos (a x) a$

Step 1.2.3.2

Reorder the factors of $\cos (a x) a$ .

$f' (x) = a \cos (a x)$

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

To apply the Chain Rule, set $u$ as $a x$ .

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

Step 2.2.2

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

$a (- \sin (u) \frac{d}{d x} [a x])$

Step 2.2.3

Replace all occurrences of $u$ with $a x$ .

$a (- \sin (a x) \frac{d}{d x} [a x])$

Step 2.3

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

$a (- \sin (a x) (a \frac{d}{d x} [x]))$

Step 2.4

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

$a^{1} a (- \sin (a x) (\frac{d}{d x} [x]))$

Step 2.5

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

$a^{1} a^{1} (- \sin (a x) (\frac{d}{d x} [x]))$

Step 2.6

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

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

Step 2.7

Add $1$ and $1$ .

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

Step 2.8

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

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

Step 2.9

Simplify the expression.

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Step 2.9.1

Multiply $- 1$ by $1$ .

$a^{2} (- \sin (a x))$

Step 2.9.2

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

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

Step 3

Find the third derivative.

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Step 3.1

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

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

Step 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 $g (x) = a x$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $a x$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $a x$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Add $1$ and $2$ .

$- a^{3} (\cos (a x) (\frac{d}{d x} [x]))$

Step 3.7

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

$- a^{3} (\cos (a x) \cdot 1)$

Step 3.8

Multiply $\cos (a x)$ by $1$ .

$f''' (x) = - a^{3} \cos (a x)$

Step 4

Find the fourth derivative.

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Step 4.1

Since $- a^{3}$ is constant with respect to $x$ , the derivative of $- a^{3} \cos (a x)$ with respect to $x$ is $- a^{3} \frac{d}{d x} [\cos (a x)]$ .

$- a^{3} \frac{d}{d x} [\cos (a x)]$

Step 4.2

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 $g (x) = a x$ .

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Step 4.2.1

To apply the Chain Rule, set $u$ as $a x$ .

$- a^{3} (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [a x])$

Step 4.2.2

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

$- a^{3} (- \sin (u) \frac{d}{d x} [a x])$

Step 4.2.3

Replace all occurrences of $u$ with $a x$ .

$- a^{3} (- \sin (a x) \frac{d}{d x} [a x])$

Step 4.3

Differentiate using the Constant Multiple Rule.

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Step 4.3.1

Multiply $- 1$ by $- 1$ .

$1 a^{3} (\sin (a x) \frac{d}{d x} [a x])$

Step 4.3.2

Multiply $a^{3}$ by $1$ .

$a^{3} (\sin (a x) \frac{d}{d x} [a x])$

Step 4.3.3

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

$a^{3} (\sin (a x) (a \frac{d}{d x} [x]))$

Step 4.4

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

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Step 4.4.1

Move $a$ .

$a \cdot a^{3} (\sin (a x) (\frac{d}{d x} [x]))$

Step 4.4.2

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

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Step 4.4.2.1

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

$a^{1} a^{3} (\sin (a x) (\frac{d}{d x} [x]))$

Step 4.4.2.2

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

$a^{1 + 3} (\sin (a x) (\frac{d}{d x} [x]))$

Step 4.4.3

Add $1$ and $3$ .

$a^{4} (\sin (a x) (\frac{d}{d x} [x]))$

Step 4.5

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

$a^{4} (\sin (a x) \cdot 1)$

Step 4.6

Multiply $\sin (a x)$ by $1$ .

$f^{4} (x) = a^{4} \sin (a x)$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $a^{4} \sin (a x)$ .

$a^{4} \sin (a x)$