Calculus Examples

$f (x) = 2 \sin^{3} (x) + 3 \sin (x) - 1$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $2 \sin^{3} (x) + 3 \sin (x) - 1$ with respect to $x$ is $\frac{d}{d x} [2 \sin^{3} (x)] + \frac{d}{d x} [3 \sin (x)] + \frac{d}{d x} [- 1]$ .

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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 $g (x) = \sin (x)$ .

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Step 1.2.2.1

To apply the Chain Rule, set $u$ as $\sin (x)$ .

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

Step 1.2.2.2

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

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

Step 1.2.2.3

Replace all occurrences of $u$ with $\sin (x)$ .

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

Step 1.2.3

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

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

Step 1.2.4

Multiply $3$ by $2$ .

$6 \sin^{2} (x) \cos (x) + \frac{d}{d x} [3 \sin (x)] + \frac{d}{d x} [- 1]$

Step 1.3

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

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Step 1.3.1

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

$6 \sin^{2} (x) \cos (x) + 3 \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- 1]$

Step 1.3.2

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

$6 \sin^{2} (x) \cos (x) + 3 \cos (x) + \frac{d}{d x} [- 1]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$6 \sin^{2} (x) \cos (x) + 3 \cos (x) + 0$

Step 1.4.2

Add $6 \sin^{2} (x) \cos (x) + 3 \cos (x)$ and $0$ .

$f' (x) = 6 \sin^{2} (x) \cos (x) + 3 \cos (x)$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $6 \sin^{2} (x) \cos (x) + 3 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [6 \sin^{2} (x) \cos (x)] + \frac{d}{d x} [3 \cos (x)]$ .

$\frac{d}{d x} [6 \sin^{2} (x) \cos (x)] + \frac{d}{d x} [3 \cos (x)]$

Step 2.2

Evaluate $\frac{d}{d x} [6 \sin^{2} (x) \cos (x)]$ .

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Step 2.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 \sin^{2} (x) \cos (x)$ with respect to $x$ is $6 \frac{d}{d x} [\sin^{2} (x) \cos (x)]$ .

$6 \frac{d}{d x} [\sin^{2} (x) \cos (x)] + \frac{d}{d x} [3 \cos (x)]$

Step 2.2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sin^{2} (x)$ and $g (x) = \cos (x)$ .

$6 (\sin^{2} (x) \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [\sin^{2} (x)]) + \frac{d}{d x} [3 \cos (x)]$

Step 2.2.3

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

$6 (\sin^{2} (x) (- \sin (x)) + \cos (x) \frac{d}{d x} [\sin^{2} (x)]) + \frac{d}{d x} [3 \cos (x)]$

Step 2.2.4

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

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Step 2.2.4.1

To apply the Chain Rule, set $u$ as $\sin (x)$ .

$6 (\sin^{2} (x) (- \sin (x)) + \cos (x) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [\sin (x)])) + \frac{d}{d x} [3 \cos (x)]$

Step 2.2.4.2

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

$6 (\sin^{2} (x) (- \sin (x)) + \cos (x) (2 u \frac{d}{d x} [\sin (x)])) + \frac{d}{d x} [3 \cos (x)]$

Step 2.2.4.3

Replace all occurrences of $u$ with $\sin (x)$ .

$6 (\sin^{2} (x) (- \sin (x)) + \cos (x) (2 \sin (x) \frac{d}{d x} [\sin (x)])) + \frac{d}{d x} [3 \cos (x)]$

Step 2.2.5

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

$6 (\sin^{2} (x) (- \sin (x)) + \cos (x) (2 \sin (x) \cos (x))) + \frac{d}{d x} [3 \cos (x)]$

Step 2.2.6

Multiply $\sin^{2} (x)$ by $\sin (x)$ by adding the exponents.

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Step 2.2.6.1

Move $\sin (x)$ .

$6 (\sin (x) \sin^{2} (x) \cdot - 1 + \cos (x) (2 \sin (x) \cos (x))) + \frac{d}{d x} [3 \cos (x)]$

Step 2.2.6.2

Multiply $\sin (x)$ by $\sin^{2} (x)$ .

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Step 2.2.6.2.1

Raise $\sin (x)$ to the power of $1$ .

$6 (\sin^{1} (x) \sin^{2} (x) \cdot - 1 + \cos (x) (2 \sin (x) \cos (x))) + \frac{d}{d x} [3 \cos (x)]$

Step 2.2.6.2.2

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

$6 ({\sin (x)}^{1 + 2} \cdot - 1 + \cos (x) (2 \sin (x) \cos (x))) + \frac{d}{d x} [3 \cos (x)]$

Step 2.2.6.3

Add $1$ and $2$ .

$6 (\sin^{3} (x) \cdot - 1 + \cos (x) (2 \sin (x) \cos (x))) + \frac{d}{d x} [3 \cos (x)]$

Step 2.2.7

Move $- 1$ to the left of $\sin^{3} (x)$ .

$6 (- 1 \cdot \sin^{3} (x) + \cos (x) (2 \sin (x) \cos (x))) + \frac{d}{d x} [3 \cos (x)]$

Step 2.2.8

Rewrite $- 1 \sin^{3} (x)$ as $- \sin^{3} (x)$ .

$6 (- \sin^{3} (x) + \cos (x) (2 \sin (x) \cos (x))) + \frac{d}{d x} [3 \cos (x)]$

Step 2.2.9

Raise $\cos (x)$ to the power of $1$ .

$6 (- \sin^{3} (x) + 2 \sin (x) (\cos^{1} (x) \cos (x))) + \frac{d}{d x} [3 \cos (x)]$

Step 2.2.10

Raise $\cos (x)$ to the power of $1$ .

$6 (- \sin^{3} (x) + 2 \sin (x) (\cos^{1} (x) \cos^{1} (x))) + \frac{d}{d x} [3 \cos (x)]$

Step 2.2.11

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

$6 (- \sin^{3} (x) + 2 \sin (x) {\cos (x)}^{1 + 1}) + \frac{d}{d x} [3 \cos (x)]$

Step 2.2.12

Add $1$ and $1$ .

$6 (- \sin^{3} (x) + 2 \sin (x) \cos^{2} (x)) + \frac{d}{d x} [3 \cos (x)]$

Step 2.3

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

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Step 2.3.1

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

$6 (- \sin^{3} (x) + 2 \sin (x) \cos^{2} (x)) + 3 \frac{d}{d x} [\cos (x)]$

Step 2.3.2

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

$6 (- \sin^{3} (x) + 2 \sin (x) \cos^{2} (x)) + 3 (- \sin (x))$

Step 2.3.3

Multiply $- 1$ by $3$ .

$6 (- \sin^{3} (x) + 2 \sin (x) \cos^{2} (x)) - 3 \sin (x)$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$6 (- \sin^{3} (x)) + 6 (2 \sin (x) \cos^{2} (x)) - 3 \sin (x)$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Multiply $- 1$ by $6$ .

$- 6 \sin^{3} (x) + 6 (2 \sin (x) \cos^{2} (x)) - 3 \sin (x)$

Step 2.4.2.2

Multiply $2$ by $6$ .

$- 6 \sin^{3} (x) + 12 \sin (x) \cos^{2} (x) - 3 \sin (x)$

Step 2.4.3

Reorder terms.

$f'' (x) = 12 \cos^{2} (x) \sin (x) - 6 \sin^{3} (x) - 3 \sin (x)$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $12 \cos^{2} (x) \sin (x) - 6 \sin^{3} (x) - 3 \sin (x)$ .

$12 \cos^{2} (x) \sin (x) - 6 \sin^{3} (x) - 3 \sin (x)$