Calculus Examples

$f (x) = 5 \sin (x) (3 x^{2} + 2 x + 1)$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $5 \sin (x) (3 x^{2} + 2 x + 1)$ with respect to $x$ is $5 \frac{d}{d x} [\sin (x) (3 x^{2} + 2 x + 1)]$ .

$5 \frac{d}{d x} [\sin (x) (3 x^{2} + 2 x + 1)]$

Step 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 $f (x) = \sin (x)$ and $g (x) = 3 x^{2} + 2 x + 1$ .

$5 (\sin (x) \frac{d}{d x} [3 x^{2} + 2 x + 1] + (3 x^{2} + 2 x + 1) \frac{d}{d x} [\sin (x)])$

Step 3

Differentiate.

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Step 3.1

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

$5 (\sin (x) (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [1]) + (3 x^{2} + 2 x + 1) \frac{d}{d x} [\sin (x)])$

Step 3.2

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

$5 (\sin (x) (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [1]) + (3 x^{2} + 2 x + 1) \frac{d}{d x} [\sin (x)])$

Step 3.3

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

$5 (\sin (x) (3 (2 x) + \frac{d}{d x} [2 x] + \frac{d}{d x} [1]) + (3 x^{2} + 2 x + 1) \frac{d}{d x} [\sin (x)])$

Step 3.4

Multiply $2$ by $3$ .

$5 (\sin (x) (6 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [1]) + (3 x^{2} + 2 x + 1) \frac{d}{d x} [\sin (x)])$

Step 3.5

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

$5 (\sin (x) (6 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [1]) + (3 x^{2} + 2 x + 1) \frac{d}{d x} [\sin (x)])$

Step 3.6

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

$5 (\sin (x) (6 x + 2 \cdot 1 + \frac{d}{d x} [1]) + (3 x^{2} + 2 x + 1) \frac{d}{d x} [\sin (x)])$

Step 3.7

Multiply $2$ by $1$ .

$5 (\sin (x) (6 x + 2 + \frac{d}{d x} [1]) + (3 x^{2} + 2 x + 1) \frac{d}{d x} [\sin (x)])$

Step 3.8

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

$5 (\sin (x) (6 x + 2 + 0) + (3 x^{2} + 2 x + 1) \frac{d}{d x} [\sin (x)])$

Step 3.9

Add $6 x + 2$ and $0$ .

$5 (\sin (x) (6 x + 2) + (3 x^{2} + 2 x + 1) \frac{d}{d x} [\sin (x)])$

Step 4

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

$5 (\sin (x) (6 x + 2) + (3 x^{2} + 2 x + 1) \cos (x))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$5 (\sin (x) (6 x) + \sin (x) \cdot 2 + (3 x^{2} + 2 x + 1) \cos (x))$

Step 5.2

Apply the distributive property.

$5 (\sin (x) (6 x) + \sin (x) \cdot 2 + 3 x^{2} \cos (x) + 2 x \cos (x) + 1 \cos (x))$

Step 5.3

Apply the distributive property.

$5 (\sin (x) (6 x)) + 5 (\sin (x) \cdot 2) + 5 (3 x^{2} \cos (x)) + 5 (2 x \cos (x)) + 5 (1 \cos (x))$

Step 5.4

Combine terms.

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Step 5.4.1

Multiply $6$ by $5$ .

$30 (\sin (x) (x)) + 5 (\sin (x) \cdot 2) + 5 (3 x^{2} \cos (x)) + 5 (2 x \cos (x)) + 5 (1 \cos (x))$

Step 5.4.2

Move $2$ to the left of $\sin (x)$ .

$30 \sin (x) x + 5 (2 \cdot \sin (x)) + 5 (3 x^{2} \cos (x)) + 5 (2 x \cos (x)) + 5 (1 \cos (x))$

Step 5.4.3

Multiply $2$ by $5$ .

$30 \sin (x) x + 10 \sin (x) + 5 (3 x^{2} \cos (x)) + 5 (2 x \cos (x)) + 5 (1 \cos (x))$

Step 5.4.4

Multiply $3$ by $5$ .

$30 \sin (x) x + 10 \sin (x) + 15 (x^{2} \cos (x)) + 5 (2 x \cos (x)) + 5 (1 \cos (x))$

Step 5.4.5

Multiply $2$ by $5$ .

$30 \sin (x) x + 10 \sin (x) + 15 x^{2} \cos (x) + 10 (x \cos (x)) + 5 (1 \cos (x))$

Step 5.4.6

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

$30 \sin (x) x + 10 \sin (x) + 15 x^{2} \cos (x) + 10 x \cos (x) + 5 \cos (x)$

Step 5.5

Reorder terms.

$30 x \sin (x) + 15 x^{2} \cos (x) + 10 x \cos (x) + 10 \sin (x) + 5 \cos (x)$