Calculus Examples

$y = \frac{x \sin (x)}{3 + \cos (x)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{x \sin (x)}{3 + \cos (x)})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x \sin (x)$ and $g (x) = 3 + \cos (x)$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Differentiate.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.4.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Step 3.4.5

Add $0$ and $\frac{d}{d x} [\cos (x)]$ .

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

Step 3.5

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

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

Step 3.6

Multiply.

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Step 3.6.1

Multiply $- 1$ by $- 1$ .

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

Step 3.6.2

Multiply $x$ by $1$ .

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Add $1$ and $1$ .

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

Step 3.11

Simplify.

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Step 3.11.1

Simplify the numerator.

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Step 3.11.1.1

Simplify each term.

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Step 3.11.1.1.1

Expand $(3 + \cos (x)) (x \cos (x) + \sin (x))$ using the FOIL Method.

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Step 3.11.1.1.1.1

Apply the distributive property.

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

Step 3.11.1.1.1.2

Apply the distributive property.

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

Step 3.11.1.1.1.3

Apply the distributive property.

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

Step 3.11.1.1.2

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

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Step 3.11.1.1.2.1

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

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

Step 3.11.1.1.2.2

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

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

Step 3.11.1.1.2.3

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

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

Step 3.11.1.1.2.4

Add $1$ and $1$ .

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

Step 3.11.1.2

Move $x \sin^{2} (x)$ .

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

Step 3.11.1.3

Factor $x$ out of $x \cos^{2} (x)$ .

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

Step 3.11.1.4

Factor $x$ out of $x \sin^{2} (x)$ .

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

Step 3.11.1.5

Factor $x$ out of $x (\cos^{2} (x)) + x (\sin^{2} (x))$ .

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

Step 3.11.1.6

Rearrange terms.

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

Step 3.11.1.7

Apply pythagorean identity.

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

Step 3.11.1.8

Multiply $x$ by $1$ .

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

Step 3.11.2

Reorder terms.

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{3 x \cos (x) + \cos (x) \sin (x) + x + 3 \sin (x)}{{(3 + \cos (x))}^{2}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{3 x \cos (x) + \cos (x) \sin (x) + x + 3 \sin (x)}{{(3 + \cos (x))}^{2}}$