Calculus Examples

$3 x \sin (x) \cos (x)$

Step 1

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

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

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 $f (x) = x \sin (x)$ and $g (x) = \cos (x)$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Add $1$ and $1$ .

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

Step 8

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

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

Step 9

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

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

Step 10

Differentiate using the Power Rule.

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Step 10.1

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

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

Step 10.2

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

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

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

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

Step 11.2

Apply the distributive property.

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

Step 11.3

Combine terms.

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Step 11.3.1

Multiply $- 1$ by $3$ .

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

Step 11.3.2

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

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

Step 11.3.3

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

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

Step 11.3.4

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

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

Step 11.3.5

Add $1$ and $1$ .

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