Calculus Examples

$y = \cos (2 x)$

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

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

Multiply $2$ by $- 1$ .

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

Step 1.2.3

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

$f' (x) = - 2 \sin (2 x) \cdot 1$

Step 1.2.4

Multiply $- 2$ by $1$ .

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

Step 2

Find the second derivative.

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Step 2.1

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

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

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 2.3

Differentiate.

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Step 2.3.1

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

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

Step 2.3.2

Multiply $2$ by $- 2$ .

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

Step 2.3.3

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

$f'' (x) = - 4 \cos (2 x) \cdot 1$

Step 2.3.4

Multiply $- 4$ by $1$ .

$f'' (x) = - 4 \cos (2 x)$

Step 3

Find the third derivative.

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Step 3.1

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

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

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

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Step 3.2.1

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

$f''' (x) = - 4 (\frac{d}{d u} (\cos (u)) \frac{d}{d x} (2 x))$

Step 3.2.2

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

$f''' (x) = - 4 (- \sin (u) \frac{d}{d x} (2 x))$

Step 3.2.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 3.3

Differentiate.

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Step 3.3.1

Multiply $- 1$ by $- 4$ .

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

Step 3.3.2

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

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

Step 3.3.3

Multiply $2$ by $4$ .

$f''' (x) = 8 \sin (2 x) \frac{d}{d x} (x)$

Step 3.3.4

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

$f''' (x) = 8 \sin (2 x) \cdot 1$

Step 3.3.5

Multiply $8$ by $1$ .

$f''' (x) = 8 \sin (2 x)$

Step 4

Find the fourth derivative.

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Step 4.1

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

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

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

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 4.3

Differentiate.

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Step 4.3.1

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

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

Step 4.3.2

Multiply $2$ by $8$ .

$f^{4} (x) = 16 \cos (2 x) \frac{d}{d x} (x)$

Step 4.3.3

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

$f^{4} (x) = 16 \cos (2 x) \cdot 1$

Step 4.3.4

Multiply $16$ by $1$ .

$f^{4} (x) = 16 \cos (2 x)$