Calculus Examples

$y = 3 \cos (x) + {(x + 2)}^{4}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $3 \cos (x) + {(x + 2)}^{4}$ with respect to $x$ is $\frac{d}{d x} [3 \cos (x)] + \frac{d}{d x} [{(x + 2)}^{4}]$ .

$\frac{d}{d x} [3 \cos (x)] + \frac{d}{d x} [{(x + 2)}^{4}]$

Step 1.2

Evaluate $\frac{d}{d x} [3 \cos (x)]$ .

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Step 1.2.1

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

$3 \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [{(x + 2)}^{4}]$

Step 1.2.2

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

$3 (- \sin (x)) + \frac{d}{d x} [{(x + 2)}^{4}]$

Step 1.2.3

Multiply $- 1$ by $3$ .

$- 3 \sin (x) + \frac{d}{d x} [{(x + 2)}^{4}]$

Step 1.3

Evaluate $\frac{d}{d x} [{(x + 2)}^{4}]$ .

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Step 1.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = x + 2$ .

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Step 1.3.1.1

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

$- 3 \sin (x) + \frac{d}{d u} [u^{4}] \frac{d}{d x} [x + 2]$

Step 1.3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

$- 3 \sin (x) + 4 u^{3} \frac{d}{d x} [x + 2]$

Step 1.3.1.3

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

$- 3 \sin (x) + 4 {(x + 2)}^{3} \frac{d}{d x} [x + 2]$

Step 1.3.2

By the Sum Rule, the derivative of $x + 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

$- 3 \sin (x) + 4 {(x + 2)}^{3} (\frac{d}{d x} [x] + \frac{d}{d x} [2])$

Step 1.3.3

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

$- 3 \sin (x) + 4 {(x + 2)}^{3} (1 + \frac{d}{d x} [2])$

Step 1.3.4

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

$- 3 \sin (x) + 4 {(x + 2)}^{3} (1 + 0)$

Step 1.3.5

Add $1$ and $0$ .

$- 3 \sin (x) + 4 {(x + 2)}^{3} \cdot 1$

Step 1.3.6

Multiply $4$ by $1$ .

$- 3 \sin (x) + 4 {(x + 2)}^{3}$

Step 1.4

Reorder terms.

$f' (x) = 4 {(x + 2)}^{3} - 3 \sin (x)$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $4 {(x + 2)}^{3} - 3 \sin (x)$ with respect to $x$ is $\frac{d}{d x} [4 {(x + 2)}^{3}] + \frac{d}{d x} [- 3 \sin (x)]$ .

$\frac{d}{d x} [4 {(x + 2)}^{3}] + \frac{d}{d x} [- 3 \sin (x)]$

Step 2.2

Evaluate $\frac{d}{d x} [4 {(x + 2)}^{3}]$ .

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Step 2.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 {(x + 2)}^{3}$ with respect to $x$ is $4 \frac{d}{d x} [{(x + 2)}^{3}]$ .

$4 \frac{d}{d x} [{(x + 2)}^{3}] + \frac{d}{d x} [- 3 \sin (x)]$

Step 2.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x + 2$ .

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Step 2.2.2.1

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

$4 (\frac{d}{d u} [u^{3}] \frac{d}{d x} [x + 2]) + \frac{d}{d x} [- 3 \sin (x)]$

Step 2.2.2.2

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

$4 (3 u^{2} \frac{d}{d x} [x + 2]) + \frac{d}{d x} [- 3 \sin (x)]$

Step 2.2.2.3

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

$4 (3 {(x + 2)}^{2} \frac{d}{d x} [x + 2]) + \frac{d}{d x} [- 3 \sin (x)]$

Step 2.2.3

By the Sum Rule, the derivative of $x + 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

$4 (3 {(x + 2)}^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [2])) + \frac{d}{d x} [- 3 \sin (x)]$

Step 2.2.4

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

$4 (3 {(x + 2)}^{2} (1 + \frac{d}{d x} [2])) + \frac{d}{d x} [- 3 \sin (x)]$

Step 2.2.5

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

$4 (3 {(x + 2)}^{2} (1 + 0)) + \frac{d}{d x} [- 3 \sin (x)]$

Step 2.2.6

Add $1$ and $0$ .

$4 (3 {(x + 2)}^{2} \cdot 1) + \frac{d}{d x} [- 3 \sin (x)]$

Step 2.2.7

Multiply $3$ by $1$ .

$4 (3 {(x + 2)}^{2}) + \frac{d}{d x} [- 3 \sin (x)]$

Step 2.2.8

Multiply $3$ by $4$ .

$12 {(x + 2)}^{2} + \frac{d}{d x} [- 3 \sin (x)]$

Step 2.3

Evaluate $\frac{d}{d x} [- 3 \sin (x)]$ .

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Step 2.3.1

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

$12 {(x + 2)}^{2} - 3 \frac{d}{d x} [\sin (x)]$

Step 2.3.2

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

$f'' (x) = 12 {(x + 2)}^{2} - 3 \cos (x)$

Step 3

Find the third derivative.

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Step 3.1

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$\frac{d}{d x} [12 ((x + 2) (x + 2)) - 3 \cos (x)]$

Step 3.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$\frac{d}{d x} [12 (x (x + 2) + 2 (x + 2)) - 3 \cos (x)]$

Step 3.2.2

Apply the distributive property.

$\frac{d}{d x} [12 (x \cdot x + x \cdot 2 + 2 (x + 2)) - 3 \cos (x)]$

Step 3.2.3

Apply the distributive property.

$\frac{d}{d x} [12 (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2) - 3 \cos (x)]$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $x$ by $x$ .

$\frac{d}{d x} [12 (x^{2} + x \cdot 2 + 2 x + 2 \cdot 2) - 3 \cos (x)]$

Step 3.3.1.2

Move $2$ to the left of $x$ .

$\frac{d}{d x} [12 (x^{2} + 2 \cdot x + 2 x + 2 \cdot 2) - 3 \cos (x)]$

Step 3.3.1.3

Multiply $2$ by $2$ .

$\frac{d}{d x} [12 (x^{2} + 2 x + 2 x + 4) - 3 \cos (x)]$

Step 3.3.2

Add $2 x$ and $2 x$ .

$\frac{d}{d x} [12 (x^{2} + 4 x + 4) - 3 \cos (x)]$

Step 3.4

By the Sum Rule, the derivative of $12 (x^{2} + 4 x + 4) - 3 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [12 (x^{2} + 4 x + 4)] + \frac{d}{d x} [- 3 \cos (x)]$ .

$\frac{d}{d x} [12 (x^{2} + 4 x + 4)] + \frac{d}{d x} [- 3 \cos (x)]$

Step 3.5

Evaluate $\frac{d}{d x} [12 (x^{2} + 4 x + 4)]$ .

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Step 3.5.1

Since $12$ is constant with respect to $x$ , the derivative of $12 (x^{2} + 4 x + 4)$ with respect to $x$ is $12 \frac{d}{d x} [x^{2} + 4 x + 4]$ .

$12 \frac{d}{d x} [x^{2} + 4 x + 4] + \frac{d}{d x} [- 3 \cos (x)]$

Step 3.5.2

By the Sum Rule, the derivative of $x^{2} + 4 x + 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [4]$ .

$12 (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [4]) + \frac{d}{d x} [- 3 \cos (x)]$

Step 3.5.3

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

$12 (2 x + \frac{d}{d x} [4 x] + \frac{d}{d x} [4]) + \frac{d}{d x} [- 3 \cos (x)]$

Step 3.5.4

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

$12 (2 x + 4 \frac{d}{d x} [x] + \frac{d}{d x} [4]) + \frac{d}{d x} [- 3 \cos (x)]$

Step 3.5.5

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

$12 (2 x + 4 \cdot 1 + \frac{d}{d x} [4]) + \frac{d}{d x} [- 3 \cos (x)]$

Step 3.5.6

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$12 (2 x + 4 \cdot 1 + 0) + \frac{d}{d x} [- 3 \cos (x)]$

Step 3.5.7

Multiply $4$ by $1$ .

$12 (2 x + 4 + 0) + \frac{d}{d x} [- 3 \cos (x)]$

Step 3.5.8

Add $2 x + 4$ and $0$ .

$12 (2 x + 4) + \frac{d}{d x} [- 3 \cos (x)]$

Step 3.6

Evaluate $\frac{d}{d x} [- 3 \cos (x)]$ .

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Step 3.6.1

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

$12 (2 x + 4) - 3 \frac{d}{d x} [\cos (x)]$

Step 3.6.2

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

$12 (2 x + 4) - 3 (- \sin (x))$

Step 3.6.3

Multiply $- 1$ by $- 3$ .

$12 (2 x + 4) + 3 \sin (x)$

Step 3.7

Simplify.

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Step 3.7.1

Apply the distributive property.

$12 (2 x) + 12 \cdot 4 + 3 \sin (x)$

Step 3.7.2

Combine terms.

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Step 3.7.2.1

Multiply $2$ by $12$ .

$24 x + 12 \cdot 4 + 3 \sin (x)$

Step 3.7.2.2

Multiply $12$ by $4$ .

$f''' (x) = 24 x + 48 + 3 \sin (x)$