Calculus Examples

$y \cos (\frac{1}{y}) = 9 x + 9 y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y \cos (\frac{1}{y})) = \frac{d}{d x} (9 x + 9 y)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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 $f (x) = y$ and $g (x) = \cos (\frac{1}{y})$ .

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

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = \frac{1}{y}$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $\frac{1}{y}$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $\frac{1}{y}$ .

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

Step 2.3

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 $f (x) = 1$ and $g (x) = y$ .

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

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Multiply $- 1$ by $1$ .

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

Step 2.4.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 2.4.3

Simplify the expression.

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Step 2.4.3.1

Multiply $y$ by $0$ .

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

Step 2.4.3.2

Subtract $\frac{d}{d x} [y]$ from $0$ .

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

Step 2.4.3.3

Move the negative in front of the fraction.

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

Step 2.4.3.4

Multiply $- 1$ by $- 1$ .

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

Step 2.4.3.5

Multiply $\sin (\frac{1}{y})$ by $1$ .

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

Step 2.5

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

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

Step 2.6

Combine $\sin (\frac{1}{y})$ and $\frac{y'}{y^{2}}$ .

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

Step 2.7

Combine $y$ and $\frac{\sin (\frac{1}{y}) y'}{y^{2}}$ .

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

Step 2.8

Cancel the common factors.

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Step 2.8.1

Factor $y$ out of $y^{2}$ .

$\frac{y (\sin (\frac{1}{y}) y')}{y \cdot y} + \cos (\frac{1}{y}) \frac{d}{d x} [y]$

Step 2.8.2

Cancel the common factor.

$\frac{y (\sin (\frac{1}{y}) y')}{y \cdot y} + \cos (\frac{1}{y}) \frac{d}{d x} [y]$

Step 2.8.3

Rewrite the expression.

$\frac{\sin (\frac{1}{y}) y'}{y} + \cos (\frac{1}{y}) \frac{d}{d x} [y]$

Step 2.9

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

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

Step 2.10

Reorder terms.

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

$\frac{d}{d x} [9 x] + \frac{d}{d x} [9 y]$

Step 3.2

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

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Step 3.2.1

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

$9 \frac{d}{d x} [x] + \frac{d}{d x} [9 y]$

Step 3.2.2

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

$9 \cdot 1 + \frac{d}{d x} [9 y]$

Step 3.2.3

Multiply $9$ by $1$ .

$9 + \frac{d}{d x} [9 y]$

Step 3.3

Evaluate $\frac{d}{d x} [9 y]$ .

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Step 3.3.1

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

$9 + 9 \frac{d}{d x} [y]$

Step 3.3.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$9 + 9 y'$

Step 3.4

Reorder terms.

$9 y' + 9$

Step 4

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

$\cos (\frac{1}{y}) y' + \frac{\sin (\frac{1}{y}) y'}{y} = 9 y' + 9$

Step 5

Solve for $y'$ .

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Step 5.1

Find the LCD of the terms in the equation.

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Step 5.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, y, 1, 1$

Step 5.1.2

The LCM of one and any expression is the expression.

$y$

Step 5.2

Multiply each term in $\cos (\frac{1}{y}) y' + \frac{\sin (\frac{1}{y}) y'}{y} = 9 y' + 9$ by $y$ to eliminate the fractions.

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Step 5.2.1

Multiply each term in $\cos (\frac{1}{y}) y' + \frac{\sin (\frac{1}{y}) y'}{y} = 9 y' + 9$ by $y$ .

$\cos (\frac{1}{y}) y' y + \frac{\sin (\frac{1}{y}) y'}{y} y = 9 y' y + 9 y$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $y$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\cos (\frac{1}{y}) y' y + \frac{\sin (\frac{1}{y}) y'}{y} y = 9 y' y + 9 y$

Step 5.2.2.1.2

Rewrite the expression.

$\cos (\frac{1}{y}) y' y + \sin (\frac{1}{y}) y' = 9 y' y + 9 y$

Step 5.2.2.2

Reorder factors in $\cos (\frac{1}{y}) y' y + \sin (\frac{1}{y}) y'$ .

$y' y \cos (\frac{1}{y}) + y' \sin (\frac{1}{y}) = 9 y' y + 9 y$

Step 5.3

Solve the equation.

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Step 5.3.1

Subtract $9 y' y$ from both sides of the equation.

$y' y \cos (\frac{1}{y}) + y' \sin (\frac{1}{y}) - 9 y' y = 9 y$

Step 5.3.2

Factor $y'$ out of $y' y \cos (\frac{1}{y}) + y' \sin (\frac{1}{y}) - 9 y' y$ .

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Step 5.3.2.1

Factor $y'$ out of $y' y \cos (\frac{1}{y})$ .

$y' (y \cos (\frac{1}{y})) + y' (\sin (\frac{1}{y})) - 9 y' y = 9 y$

Step 5.3.2.2

Factor $y'$ out of $y' \sin (\frac{1}{y})$ .

$y' (y \cos (\frac{1}{y})) + y' (\sin (\frac{1}{y})) - 9 y' y = 9 y$

Step 5.3.2.3

Factor $y'$ out of $- 9 y' y$ .

$y' (y \cos (\frac{1}{y})) + y' (\sin (\frac{1}{y})) + y' (- 9 y) = 9 y$

Step 5.3.2.4

Factor $y'$ out of $y' (y \cos (\frac{1}{y})) + y' (\sin (\frac{1}{y}))$ .

$y' (y \cos (\frac{1}{y}) + \sin (\frac{1}{y})) + y' (- 9 y) = 9 y$

Step 5.3.2.5

Factor $y'$ out of $y' (y \cos (\frac{1}{y}) + \sin (\frac{1}{y})) + y' (- 9 y)$ .

$y' (y \cos (\frac{1}{y}) + \sin (\frac{1}{y}) - 9 y) = 9 y$

Step 5.3.3

Divide each term in $y' (y \cos (\frac{1}{y}) + \sin (\frac{1}{y}) - 9 y) = 9 y$ by $y \cos (\frac{1}{y}) + \sin (\frac{1}{y}) - 9 y$ and simplify.

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Step 5.3.3.1

Divide each term in $y' (y \cos (\frac{1}{y}) + \sin (\frac{1}{y}) - 9 y) = 9 y$ by $y \cos (\frac{1}{y}) + \sin (\frac{1}{y}) - 9 y$ .

$\frac{y' (y \cos (\frac{1}{y}) + \sin (\frac{1}{y}) - 9 y)}{y \cos (\frac{1}{y}) + \sin (\frac{1}{y}) - 9 y} = \frac{9 y}{y \cos (\frac{1}{y}) + \sin (\frac{1}{y}) - 9 y}$

Step 5.3.3.2

Simplify the left side.

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Step 5.3.3.2.1

Cancel the common factor of $y \cos (\frac{1}{y}) + \sin (\frac{1}{y}) - 9 y$ .

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Step 5.3.3.2.1.1

Cancel the common factor.

$\frac{y' (y \cos (\frac{1}{y}) + \sin (\frac{1}{y}) - 9 y)}{y \cos (\frac{1}{y}) + \sin (\frac{1}{y}) - 9 y} = \frac{9 y}{y \cos (\frac{1}{y}) + \sin (\frac{1}{y}) - 9 y}$

Step 5.3.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{9 y}{y \cos (\frac{1}{y}) + \sin (\frac{1}{y}) - 9 y}$

Step 6

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

$\frac{d y}{d x} = \frac{9 y}{y \cos (\frac{1}{y}) + \sin (\frac{1}{y}) - 9 y}$