Calculus Examples

$f (x) = \sin (x) + x^{3} - 7 x + 2$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $\sin (x) + x^{3} - 7 x + 2$ with respect to $x$ is $\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [2]$

Step 1.2

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

$\cos (x) + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [2]$

Step 1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\cos (x) + 3 x^{2} + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [2]$

Step 1.4

Evaluate $\frac{d}{d x} [- 7 x]$ .

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Step 1.4.1

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

$\cos (x) + 3 x^{2} - 7 \frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 1.4.2

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

$\cos (x) + 3 x^{2} - 7 \cdot 1 + \frac{d}{d x} [2]$

Step 1.4.3

Multiply $- 7$ by $1$ .

$\cos (x) + 3 x^{2} - 7 + \frac{d}{d x} [2]$

Step 1.5

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\cos (x) + 3 x^{2} - 7 + 0$

Step 1.6

Simplify.

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Step 1.6.1

Add $\cos (x) + 3 x^{2} - 7$ and $0$ .

$\cos (x) + 3 x^{2} - 7$

Step 1.6.2

Reorder terms.

$f' (x) = 3 x^{2} - 7 + \cos (x)$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $3 x^{2} - 7 + \cos (x)$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 7] + \frac{d}{d x} [\cos (x)]$ .

$\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 7] + \frac{d}{d x} [\cos (x)]$

Step 2.2

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 2.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 7] + \frac{d}{d x} [\cos (x)]$

Step 2.2.2

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

$3 (2 x) + \frac{d}{d x} [- 7] + \frac{d}{d x} [\cos (x)]$

Step 2.2.3

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [- 7] + \frac{d}{d x} [\cos (x)]$

Step 2.3

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

$6 x + 0 + \frac{d}{d x} [\cos (x)]$

Step 2.4

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

$6 x + 0 - \sin (x)$

Step 2.5

Add $6 x$ and $0$ .

$f'' (x) = 6 x - \sin (x)$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $6 x - \sin (x)$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [- \sin (x)]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [- \sin (x)]$

Step 3.2

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 3.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x] + \frac{d}{d x} [- \sin (x)]$

Step 3.2.2

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

$6 \cdot 1 + \frac{d}{d x} [- \sin (x)]$

Step 3.2.3

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [- \sin (x)]$

Step 3.3

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

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Step 3.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin (x)$ with respect to $x$ is $- \frac{d}{d x} [\sin (x)]$ .

$6 - \frac{d}{d x} [\sin (x)]$

Step 3.3.2

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

$f''' (x) = 6 - \cos (x)$

Step 4

Find the fourth derivative.

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Step 4.1

Differentiate.

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Step 4.1.1

By the Sum Rule, the derivative of $6 - \cos (x)$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [- \cos (x)]$ .

$\frac{d}{d x} [6] + \frac{d}{d x} [- \cos (x)]$

Step 4.1.2

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \cos (x)]$

Step 4.2

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

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Step 4.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (x)]$ .

$0 - \frac{d}{d x} [\cos (x)]$

Step 4.2.2

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

$0 - - \sin (x)$

Step 4.2.3

Multiply $- 1$ by $- 1$ .

$0 + 1 \sin (x)$

Step 4.2.4

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

$0 + \sin (x)$

Step 4.3

Add $0$ and $\sin (x)$ .

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

Step 5

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

$\sin (x)$