Calculus Examples

$\frac{7 + \sin (x)}{7 x + \cos (x)}$

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 $f (x) = 7 + \sin (x)$ and $g (x) = 7 x + \cos (x)$ .

$\frac{(7 x + \cos (x)) \frac{d}{d x} [7 + \sin (x)] - (7 + \sin (x)) \frac{d}{d x} [7 x + \cos (x)]}{{(7 x + \cos (x))}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $7 + \sin (x)$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [\sin (x)]$ .

$\frac{(7 x + \cos (x)) (\frac{d}{d x} [7] + \frac{d}{d x} [\sin (x)]) - (7 + \sin (x)) \frac{d}{d x} [7 x + \cos (x)]}{{(7 x + \cos (x))}^{2}}$

Step 2.2

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$\frac{(7 x + \cos (x)) (0 + \frac{d}{d x} [\sin (x)]) - (7 + \sin (x)) \frac{d}{d x} [7 x + \cos (x)]}{{(7 x + \cos (x))}^{2}}$

Step 2.3

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

$\frac{(7 x + \cos (x)) \frac{d}{d x} [\sin (x)] - (7 + \sin (x)) \frac{d}{d x} [7 x + \cos (x)]}{{(7 x + \cos (x))}^{2}}$

Step 3

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

$\frac{(7 x + \cos (x)) \cos (x) - (7 + \sin (x)) \frac{d}{d x} [7 x + \cos (x)]}{{(7 x + \cos (x))}^{2}}$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $7 x + \cos (x)$ with respect to $x$ is $\frac{d}{d x} [7 x] + \frac{d}{d x} [\cos (x)]$ .

$\frac{(7 x + \cos (x)) \cos (x) - (7 + \sin (x)) (\frac{d}{d x} [7 x] + \frac{d}{d x} [\cos (x)])}{{(7 x + \cos (x))}^{2}}$

Step 4.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$\frac{(7 x + \cos (x)) \cos (x) - (7 + \sin (x)) (7 \frac{d}{d x} [x] + \frac{d}{d x} [\cos (x)])}{{(7 x + \cos (x))}^{2}}$

Step 4.3

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

$\frac{(7 x + \cos (x)) \cos (x) - (7 + \sin (x)) (7 \cdot 1 + \frac{d}{d x} [\cos (x)])}{{(7 x + \cos (x))}^{2}}$

Step 4.4

Multiply $7$ by $1$ .

$\frac{(7 x + \cos (x)) \cos (x) - (7 + \sin (x)) (7 + \frac{d}{d x} [\cos (x)])}{{(7 x + \cos (x))}^{2}}$

Step 5

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

$\frac{(7 x + \cos (x)) \cos (x) - (7 + \sin (x)) (7 - \sin (x))}{{(7 x + \cos (x))}^{2}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{7 x \cos (x) + \cos (x) \cos (x) - (7 + \sin (x)) (7 - \sin (x))}{{(7 x + \cos (x))}^{2}}$

Step 6.2

Apply the distributive property.

$\frac{7 x \cos (x) + \cos (x) \cos (x) + (- 1 \cdot 7 - \sin (x)) (7 - \sin (x))}{{(7 x + \cos (x))}^{2}}$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

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

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Step 6.3.1.1.1

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

$\frac{7 x \cos (x) + \cos^{1} (x) \cos (x) + (- 1 \cdot 7 - \sin (x)) (7 - \sin (x))}{{(7 x + \cos (x))}^{2}}$

Step 6.3.1.1.2

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

$\frac{7 x \cos (x) + \cos^{1} (x) \cos^{1} (x) + (- 1 \cdot 7 - \sin (x)) (7 - \sin (x))}{{(7 x + \cos (x))}^{2}}$

Step 6.3.1.1.3

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

$\frac{7 x \cos (x) + {\cos (x)}^{1 + 1} + (- 1 \cdot 7 - \sin (x)) (7 - \sin (x))}{{(7 x + \cos (x))}^{2}}$

Step 6.3.1.1.4

Add $1$ and $1$ .

$\frac{7 x \cos (x) + \cos^{2} (x) + (- 1 \cdot 7 - \sin (x)) (7 - \sin (x))}{{(7 x + \cos (x))}^{2}}$

Step 6.3.1.2

Multiply $- 1$ by $7$ .

$\frac{7 x \cos (x) + \cos^{2} (x) + (- 7 - \sin (x)) (7 - \sin (x))}{{(7 x + \cos (x))}^{2}}$

Step 6.3.1.3

Expand $(- 7 - \sin (x)) (7 - \sin (x))$ using the FOIL Method.

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Step 6.3.1.3.1

Apply the distributive property.

$\frac{7 x \cos (x) + \cos^{2} (x) - 7 (7 - \sin (x)) - \sin (x) (7 - \sin (x))}{{(7 x + \cos (x))}^{2}}$

Step 6.3.1.3.2

Apply the distributive property.

$\frac{7 x \cos (x) + \cos^{2} (x) - 7 \cdot 7 - 7 (- \sin (x)) - \sin (x) (7 - \sin (x))}{{(7 x + \cos (x))}^{2}}$

Step 6.3.1.3.3

Apply the distributive property.

$\frac{7 x \cos (x) + \cos^{2} (x) - 7 \cdot 7 - 7 (- \sin (x)) - \sin (x) \cdot 7 - \sin (x) (- \sin (x))}{{(7 x + \cos (x))}^{2}}$

Step 6.3.1.4

Simplify and combine like terms.

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Step 6.3.1.4.1

Simplify each term.

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Step 6.3.1.4.1.1

Multiply $- 7$ by $7$ .

$\frac{7 x \cos (x) + \cos^{2} (x) - 49 - 7 (- \sin (x)) - \sin (x) \cdot 7 - \sin (x) (- \sin (x))}{{(7 x + \cos (x))}^{2}}$

Step 6.3.1.4.1.2

Multiply $- 1$ by $- 7$ .

$\frac{7 x \cos (x) + \cos^{2} (x) - 49 + 7 \sin (x) - \sin (x) \cdot 7 - \sin (x) (- \sin (x))}{{(7 x + \cos (x))}^{2}}$

Step 6.3.1.4.1.3

Multiply $7$ by $- 1$ .

$\frac{7 x \cos (x) + \cos^{2} (x) - 49 + 7 \sin (x) - 7 \sin (x) - \sin (x) (- \sin (x))}{{(7 x + \cos (x))}^{2}}$

Step 6.3.1.4.1.4

Multiply $- \sin (x) (- \sin (x))$ .

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Step 6.3.1.4.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{7 x \cos (x) + \cos^{2} (x) - 49 + 7 \sin (x) - 7 \sin (x) + 1 \sin (x) \sin (x)}{{(7 x + \cos (x))}^{2}}$

Step 6.3.1.4.1.4.2

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

$\frac{7 x \cos (x) + \cos^{2} (x) - 49 + 7 \sin (x) - 7 \sin (x) + \sin (x) \sin (x)}{{(7 x + \cos (x))}^{2}}$

Step 6.3.1.4.1.4.3

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

$\frac{7 x \cos (x) + \cos^{2} (x) - 49 + 7 \sin (x) - 7 \sin (x) + \sin^{1} (x) \sin (x)}{{(7 x + \cos (x))}^{2}}$

Step 6.3.1.4.1.4.4

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

$\frac{7 x \cos (x) + \cos^{2} (x) - 49 + 7 \sin (x) - 7 \sin (x) + \sin^{1} (x) \sin^{1} (x)}{{(7 x + \cos (x))}^{2}}$

Step 6.3.1.4.1.4.5

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

$\frac{7 x \cos (x) + \cos^{2} (x) - 49 + 7 \sin (x) - 7 \sin (x) + {\sin (x)}^{1 + 1}}{{(7 x + \cos (x))}^{2}}$

Step 6.3.1.4.1.4.6

Add $1$ and $1$ .

$\frac{7 x \cos (x) + \cos^{2} (x) - 49 + 7 \sin (x) - 7 \sin (x) + \sin^{2} (x)}{{(7 x + \cos (x))}^{2}}$

Step 6.3.1.4.2

Subtract $7 \sin (x)$ from $7 \sin (x)$ .

$\frac{7 x \cos (x) + \cos^{2} (x) - 49 + 0 + \sin^{2} (x)}{{(7 x + \cos (x))}^{2}}$

Step 6.3.1.4.3

Add $- 49$ and $0$ .

$\frac{7 x \cos (x) + \cos^{2} (x) - 49 + \sin^{2} (x)}{{(7 x + \cos (x))}^{2}}$

Step 6.3.2

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

$\frac{7 x \cos (x) + \cos^{2} (x) + \sin^{2} (x) - 49}{{(7 x + \cos (x))}^{2}}$

Step 6.3.3

Rearrange terms.

$\frac{7 x \cos (x) + \sin^{2} (x) + \cos^{2} (x) - 49}{{(7 x + \cos (x))}^{2}}$

Step 6.3.4

Apply pythagorean identity.

$\frac{7 x \cos (x) + 1 - 49}{{(7 x + \cos (x))}^{2}}$

Step 6.3.5

Subtract $49$ from $1$ .

$\frac{7 x \cos (x) - 48}{{(7 x + \cos (x))}^{2}}$