Calculus Examples

$y = \frac{\sin (x)}{9 x} + \frac{9 x}{\sin (x)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{\sin (x)}{9 x} + \frac{9 x}{\sin (x)})$

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

By the Sum Rule, the derivative of $\frac{\sin (x)}{9 x} + \frac{9 x}{\sin (x)}$ with respect to $x$ is $\frac{d}{d x} [\frac{\sin (x)}{9 x}] + \frac{d}{d x} [\frac{9 x}{\sin (x)}]$ .

$\frac{d}{d x} [\frac{\sin (x)}{9 x}] + \frac{d}{d x} [\frac{9 x}{\sin (x)}]$

Step 3.2

Evaluate $\frac{d}{d x} [\frac{\sin (x)}{9 x}]$ .

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Step 3.2.1

Since $\frac{1}{9}$ is constant with respect to $x$ , the derivative of $\frac{\sin (x)}{9 x}$ with respect to $x$ is $\frac{1}{9} \frac{d}{d x} [\frac{\sin (x)}{x}]$ .

$\frac{1}{9} \frac{d}{d x} [\frac{\sin (x)}{x}] + \frac{d}{d x} [\frac{9 x}{\sin (x)}]$

Step 3.2.2

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

$\frac{1}{9} \cdot \frac{x \frac{d}{d x} [\sin (x)] - \sin (x) \frac{d}{d x} [x]}{x^{2}} + \frac{d}{d x} [\frac{9 x}{\sin (x)}]$

Step 3.2.3

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

$\frac{1}{9} \cdot \frac{x \cos (x) - \sin (x) \frac{d}{d x} [x]}{x^{2}} + \frac{d}{d x} [\frac{9 x}{\sin (x)}]$

Step 3.2.4

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

$\frac{1}{9} \cdot \frac{x \cos (x) - \sin (x) \cdot 1}{x^{2}} + \frac{d}{d x} [\frac{9 x}{\sin (x)}]$

Step 3.2.5

Multiply $- 1$ by $1$ .

$\frac{1}{9} \cdot \frac{x \cos (x) - \sin (x)}{x^{2}} + \frac{d}{d x} [\frac{9 x}{\sin (x)}]$

Step 3.2.6

Multiply $\frac{1}{9}$ by $\frac{x \cos (x) - \sin (x)}{x^{2}}$ .

$\frac{x \cos (x) - \sin (x)}{9 x^{2}} + \frac{d}{d x} [\frac{9 x}{\sin (x)}]$

Step 3.3

Evaluate $\frac{d}{d x} [\frac{9 x}{\sin (x)}]$ .

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Step 3.3.1

Since $9$ is constant with respect to $x$ , the derivative of $\frac{9 x}{\sin (x)}$ with respect to $x$ is $9 \frac{d}{d x} [\frac{x}{\sin (x)}]$ .

$\frac{x \cos (x) - \sin (x)}{9 x^{2}} + 9 \frac{d}{d x} [\frac{x}{\sin (x)}]$

Step 3.3.2

$\frac{x \cos (x) - \sin (x)}{9 x^{2}} + 9 \frac{\sin (x) \frac{d}{d x} [x] - x \frac{d}{d x} [\sin (x)]}{\sin^{2} (x)}$

Step 3.3.3

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

$\frac{x \cos (x) - \sin (x)}{9 x^{2}} + 9 \frac{\sin (x) \cdot 1 - x \frac{d}{d x} [\sin (x)]}{\sin^{2} (x)}$

Step 3.3.4

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

$\frac{x \cos (x) - \sin (x)}{9 x^{2}} + 9 \frac{\sin (x) \cdot 1 - x \cos (x)}{\sin^{2} (x)}$

Step 3.3.5

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

$\frac{x \cos (x) - \sin (x)}{9 x^{2}} + 9 \frac{\sin (x) - x \cos (x)}{\sin^{2} (x)}$

Step 3.3.6

Combine $9$ and $\frac{\sin (x) - x \cos (x)}{\sin^{2} (x)}$ .

$\frac{x \cos (x) - \sin (x)}{9 x^{2}} + \frac{9 (\sin (x) - x \cos (x))}{\sin^{2} (x)}$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$\frac{x \cos (x) - \sin (x)}{9 x^{2}} + \frac{9 \sin (x) + 9 (- x \cos (x))}{\sin^{2} (x)}$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Multiply $- 1$ by $9$ .

$\frac{x \cos (x) - \sin (x)}{9 x^{2}} + \frac{9 \sin (x) - 9 (x \cos (x))}{\sin^{2} (x)}$

Step 3.4.2.2

To write $\frac{x \cos (x) - \sin (x)}{9 x^{2}}$ as a fraction with a common denominator, multiply by $\frac{\sin^{2} (x)}{\sin^{2} (x)}$ .

$\frac{x \cos (x) - \sin (x)}{9 x^{2}} \cdot \frac{\sin^{2} (x)}{\sin^{2} (x)} + \frac{9 \sin (x) - 9 x \cos (x)}{\sin^{2} (x)}$

Step 3.4.2.3

To write $\frac{9 \sin (x) - 9 x \cos (x)}{\sin^{2} (x)}$ as a fraction with a common denominator, multiply by $\frac{9 x^{2}}{9 x^{2}}$ .

$\frac{x \cos (x) - \sin (x)}{9 x^{2}} \cdot \frac{\sin^{2} (x)}{\sin^{2} (x)} + \frac{9 \sin (x) - 9 x \cos (x)}{\sin^{2} (x)} \cdot \frac{9 x^{2}}{9 x^{2}}$

Step 3.4.2.4

Write each expression with a common denominator of $9 x^{2} \sin^{2} (x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.2.4.1

Multiply $\frac{x \cos (x) - \sin (x)}{9 x^{2}}$ by $\frac{\sin^{2} (x)}{\sin^{2} (x)}$ .

$\frac{(x \cos (x) - \sin (x)) \sin^{2} (x)}{9 x^{2} \sin^{2} (x)} + \frac{9 \sin (x) - 9 x \cos (x)}{\sin^{2} (x)} \cdot \frac{9 x^{2}}{9 x^{2}}$

Step 3.4.2.4.2

Multiply $\frac{9 \sin (x) - 9 x \cos (x)}{\sin^{2} (x)}$ by $\frac{9 x^{2}}{9 x^{2}}$ .

$\frac{(x \cos (x) - \sin (x)) \sin^{2} (x)}{9 x^{2} \sin^{2} (x)} + \frac{(9 \sin (x) - 9 x \cos (x)) (9 x^{2})}{\sin^{2} (x) (9 x^{2})}$

Step 3.4.2.4.3

Reorder the factors of $\sin^{2} (x) (9 x^{2})$ .

$\frac{(x \cos (x) - \sin (x)) \sin^{2} (x)}{9 x^{2} \sin^{2} (x)} + \frac{(9 \sin (x) - 9 x \cos (x)) (9 x^{2})}{9 x^{2} \sin^{2} (x)}$

Step 3.4.2.5

Combine the numerators over the common denominator.

$\frac{(x \cos (x) - \sin (x)) \sin^{2} (x) + (9 \sin (x) - 9 x \cos (x)) (9 x^{2})}{9 x^{2} \sin^{2} (x)}$

Step 3.4.2.6

Move $9$ to the left of $9 \sin (x) - 9 x \cos (x)$ .

$\frac{(x \cos (x) - \sin (x)) \sin^{2} (x) + 9 (9 \sin (x) - 9 x \cos (x)) x^{2}}{9 x^{2} \sin^{2} (x)}$

Step 3.4.3

Reorder terms.

$\frac{\sin^{2} (x) (x \cos (x) - \sin (x)) + 9 x^{2} (9 \sin (x) - 9 x \cos (x))}{9 x^{2} \sin^{2} (x)}$

Step 4

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

$y' = \frac{\sin^{2} (x) (x \cos (x) - \sin (x)) + 9 x^{2} (9 \sin (x) - 9 x \cos (x))}{9 x^{2} \sin^{2} (x)}$

Step 5

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

$\frac{d y}{d x} = \frac{\sin^{2} (x) (x \cos (x) - \sin (x)) + 9 x^{2} (9 \sin (x) - 9 x \cos (x))}{9 x^{2} \sin^{2} (x)}$