Calculus Examples

y = \frac{x sin (x)}{1 + cos (x)}

$y = \frac{x \sin (x)}{1 + \cos (x)}$

Step 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

$g (x) = 1 + \cos (x)$ .

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

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

$f (x) = x$ and

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

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

Step 3

The derivative of

$\sin (x)$ with respect to

$x$ is

$\cos (x)$ .

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

Step 4

Differentiate.

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Step 4.1

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 4.2

Multiply

$\sin (x)$ by

$1$ .

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

Step 4.3

By the Sum Rule, the derivative of

$1 + \cos (x)$ with respect to

$x$ is

$\frac{d}{d x} [1] + \frac{d}{d x} [\cos (x)]$ .

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

Step 4.4

Since

$1$ is constant with respect to

$x$ , the derivative of

$1$ with respect to

$x$ is

$0$ .

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

Step 4.5

Add

$0$ and

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

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

Step 5

The derivative of

$\cos (x)$ with respect to

$x$ is

$- \sin (x)$ .

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

Step 6

Multiply.

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Step 6.1

Multiply

$- 1$ by

$- 1$ .

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

Step 6.2

Multiply

$x$ by

$1$ .

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

Step 7

Raise

$\sin (x)$ to the power of

$1$ .

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

Step 8

Raise

$\sin (x)$ to the power of

$1$ .

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

Step 9

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 10

Add

$1$ and

$1$ .

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

Step 11

Simplify.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Simplify each term.

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Step 11.1.1.1

Expand

$(1 + \cos (x)) (x \cos (x) + \sin (x))$ using the FOIL Method.

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Step 11.1.1.1.1

Apply the distributive property.

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

Step 11.1.1.1.2

Apply the distributive property.

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

Step 11.1.1.1.3

Apply the distributive property.

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

Step 11.1.1.2

Simplify each term.

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Step 11.1.1.2.1

Multiply

$x \cos (x)$ by

$1$ .

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

Step 11.1.1.2.2

Multiply

$\sin (x)$ by

$1$ .

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

Step 11.1.1.2.3

Multiply

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

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Step 11.1.1.2.3.1

Raise

$\cos (x)$ to the power of

$1$ .

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

Step 11.1.1.2.3.2

Raise

$\cos (x)$ to the power of

$1$ .

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

Step 11.1.1.2.3.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 11.1.1.2.3.4

Add

$1$ and

$1$ .

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

Step 11.1.2

Move

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

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

Step 11.1.3

Factor

$x$ out of

$x \cos^{2} (x)$ .

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

Step 11.1.4

Factor

$x$ out of

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

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

Step 11.1.5

Factor

$x$ out of

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

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

Step 11.1.6

Rearrange terms.

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

Step 11.1.7

Apply pythagorean identity.

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

Step 11.1.8

Multiply

$x$ by

$1$ .

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

Step 11.2

Reorder terms.

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

Step 11.3

Simplify the numerator.

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Step 11.3.1

Factor out the greatest common factor from each group.

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Step 11.3.1.1

Group the first two terms and the last two terms.

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

Step 11.3.1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 11.3.2

Factor the polynomial by factoring out the greatest common factor,

$x + \sin (x)$ .

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

Step 11.4

Cancel the common factor of

$\cos (x) + 1$ and

${(1 + \cos (x))}^{2}$ .

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Step 11.4.1

Reorder terms.

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

Step 11.4.2

Factor

$1 + \cos (x)$ out of

$(x + \sin (x)) (1 + \cos (x))$ .

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

Step 11.4.3

Cancel the common factors.

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Step 11.4.3.1

Factor

$1 + \cos (x)$ out of

${(1 + \cos (x))}^{2}$ .

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

Step 11.4.3.2

Cancel the common factor.

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

Step 11.4.3.3

Rewrite the expression.

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