Calculus Examples

$y = \sin (4 x - 7)$

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 $g (x) = 4 x - 7$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $4 x - 7$ .

$\frac{d}{d u} [\sin (u)] \frac{d}{d x} [4 x - 7]$

Step 1.1.2

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

$\cos (u) \frac{d}{d x} [4 x - 7]$

Step 1.1.3

Replace all occurrences of $u$ with $4 x - 7$ .

$\cos (4 x - 7) \frac{d}{d x} [4 x - 7]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $4 x - 7$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 7]$ .

$\cos (4 x - 7) (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 7])$

Step 1.2.2

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

$\cos (4 x - 7) (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 7])$

Step 1.2.3

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

$\cos (4 x - 7) (4 \cdot 1 + \frac{d}{d x} [- 7])$

Step 1.2.4

Multiply $4$ by $1$ .

$\cos (4 x - 7) (4 + \frac{d}{d x} [- 7])$

Step 1.2.5

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$\cos (4 x - 7) (4 + 0)$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $4$ and $0$ .

$\cos (4 x - 7) \cdot 4$

Step 1.2.6.2

Move $4$ to the left of $\cos (4 x - 7)$ .

$f' (x) = 4 \cos (4 x - 7)$

Step 2

Find the second derivative.

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 \cos (4 x - 7)$ with respect to $x$ is $4 \frac{d}{d x} [\cos (4 x - 7)]$ .

$4 \frac{d}{d x} [\cos (4 x - 7)]$

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 $g (x) = 4 x - 7$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $4 x - 7$ .

$4 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [4 x - 7])$

Step 2.2.2

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

$4 (- \sin (u) \frac{d}{d x} [4 x - 7])$

Step 2.2.3

Replace all occurrences of $u$ with $4 x - 7$ .

$4 (- \sin (4 x - 7) \frac{d}{d x} [4 x - 7])$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 1$ by $4$ .

$- 4 (\sin (4 x - 7) \frac{d}{d x} [4 x - 7])$

Step 2.3.2

By the Sum Rule, the derivative of $4 x - 7$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 7]$ .

$- 4 \sin (4 x - 7) (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 7])$

Step 2.3.3

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

$- 4 \sin (4 x - 7) (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 7])$

Step 2.3.4

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

$- 4 \sin (4 x - 7) (4 \cdot 1 + \frac{d}{d x} [- 7])$

Step 2.3.5

Multiply $4$ by $1$ .

$- 4 \sin (4 x - 7) (4 + \frac{d}{d x} [- 7])$

Step 2.3.6

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$- 4 \sin (4 x - 7) (4 + 0)$

Step 2.3.7

Simplify the expression.

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Step 2.3.7.1

Add $4$ and $0$ .

$- 4 \sin (4 x - 7) \cdot 4$

Step 2.3.7.2

Multiply $4$ by $- 4$ .

$f'' (x) = - 16 \sin (4 x - 7)$

Step 3

Find the third derivative.

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Step 3.1

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16 \sin (4 x - 7)$ with respect to $x$ is $- 16 \frac{d}{d x} [\sin (4 x - 7)]$ .

$- 16 \frac{d}{d x} [\sin (4 x - 7)]$

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 $g (x) = 4 x - 7$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $4 x - 7$ .

$- 16 (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [4 x - 7])$

Step 3.2.2

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

$- 16 (\cos (u) \frac{d}{d x} [4 x - 7])$

Step 3.2.3

Replace all occurrences of $u$ with $4 x - 7$ .

$- 16 (\cos (4 x - 7) \frac{d}{d x} [4 x - 7])$

Step 3.3

Differentiate.

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Step 3.3.1

By the Sum Rule, the derivative of $4 x - 7$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 7]$ .

$- 16 \cos (4 x - 7) (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 7])$

Step 3.3.2

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

$- 16 \cos (4 x - 7) (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 7])$

Step 3.3.3

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

$- 16 \cos (4 x - 7) (4 \cdot 1 + \frac{d}{d x} [- 7])$

Step 3.3.4

Multiply $4$ by $1$ .

$- 16 \cos (4 x - 7) (4 + \frac{d}{d x} [- 7])$

Step 3.3.5

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$- 16 \cos (4 x - 7) (4 + 0)$

Step 3.3.6

Simplify the expression.

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Step 3.3.6.1

Add $4$ and $0$ .

$- 16 \cos (4 x - 7) \cdot 4$

Step 3.3.6.2

Multiply $4$ by $- 16$ .

$f''' (x) = - 64 \cos (4 x - 7)$

Step 4

Find the fourth derivative.

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Step 4.1

Since $- 64$ is constant with respect to $x$ , the derivative of $- 64 \cos (4 x - 7)$ with respect to $x$ is $- 64 \frac{d}{d x} [\cos (4 x - 7)]$ .

$- 64 \frac{d}{d x} [\cos (4 x - 7)]$

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 $g (x) = 4 x - 7$ .

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Step 4.2.1

To apply the Chain Rule, set $u$ as $4 x - 7$ .

$- 64 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [4 x - 7])$

Step 4.2.2

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

$- 64 (- \sin (u) \frac{d}{d x} [4 x - 7])$

Step 4.2.3

Replace all occurrences of $u$ with $4 x - 7$ .

$- 64 (- \sin (4 x - 7) \frac{d}{d x} [4 x - 7])$

Step 4.3

Differentiate.

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Step 4.3.1

Multiply $- 1$ by $- 64$ .

$64 (\sin (4 x - 7) \frac{d}{d x} [4 x - 7])$

Step 4.3.2

By the Sum Rule, the derivative of $4 x - 7$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 7]$ .

$64 \sin (4 x - 7) (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 7])$

Step 4.3.3

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

$64 \sin (4 x - 7) (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 7])$

Step 4.3.4

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

$64 \sin (4 x - 7) (4 \cdot 1 + \frac{d}{d x} [- 7])$

Step 4.3.5

Multiply $4$ by $1$ .

$64 \sin (4 x - 7) (4 + \frac{d}{d x} [- 7])$

Step 4.3.6

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$64 \sin (4 x - 7) (4 + 0)$

Step 4.3.7

Simplify the expression.

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Step 4.3.7.1

Add $4$ and $0$ .

$64 \sin (4 x - 7) \cdot 4$

Step 4.3.7.2

Multiply $4$ by $64$ .

$f^{4} (x) = 256 \sin (4 x - 7)$