Algebra Examples

$5 \tan (\frac{x}{y}) = 12 x$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (5 \tan (\frac{x}{y})) = \frac{d}{d x} (12 x)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 \tan (\frac{x}{y})$ with respect to $x$ is $5 \frac{d}{d x} [\tan (\frac{x}{y})]$ .

$5 \frac{d}{d x} [\tan (\frac{x}{y})]$

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

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Step 2.2.1

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

$5 (\frac{d}{d u} [\tan (u)] \frac{d}{d x} [\frac{x}{y}])$

Step 2.2.2

The derivative of $\tan (u)$ with respect to $u$ is $\sec^{2} (u)$ .

$5 (\sec^{2} (u) \frac{d}{d x} [\frac{x}{y}])$

Step 2.2.3

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

$5 (\sec^{2} (\frac{x}{y}) \frac{d}{d x} [\frac{x}{y}])$

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

$5 \sec^{2} (\frac{x}{y}) \frac{y \frac{d}{d x} [x] - x \frac{d}{d x} [y]}{y^{2}}$

Step 2.4

Differentiate using the Power Rule.

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Step 2.4.1

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

$5 \sec^{2} (\frac{x}{y}) \frac{y \cdot 1 - x \frac{d}{d x} [y]}{y^{2}}$

Step 2.4.2

Multiply $y$ by $1$ .

$5 \sec^{2} (\frac{x}{y}) \frac{y - x \frac{d}{d x} [y]}{y^{2}}$

Step 2.5

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

$5 \sec^{2} (\frac{x}{y}) \frac{y - x y'}{y^{2}}$

Step 2.6

Combine $\frac{y - x y'}{y^{2}}$ and $5$ .

$\frac{(y - x y') \cdot 5}{y^{2}} \sec^{2} (\frac{x}{y})$

Step 2.7

Combine $\frac{(y - x y') \cdot 5}{y^{2}}$ and $\sec^{2} (\frac{x}{y})$ .

$\frac{(y - x y') \cdot 5 \sec^{2} (\frac{x}{y})}{y^{2}}$

Step 2.8

Move $5$ to the left of $y - x y'$ .

$\frac{5 \cdot (y - x y') \sec^{2} (\frac{x}{y})}{y^{2}}$

Step 2.9

Simplify.

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Step 2.9.1

Apply the distributive property.

$\frac{(5 y + 5 (- x y')) \sec^{2} (\frac{x}{y})}{y^{2}}$

Step 2.9.2

Apply the distributive property.

$\frac{5 y \sec^{2} (\frac{x}{y}) + 5 (- x y') \sec^{2} (\frac{x}{y})}{y^{2}}$

Step 2.9.3

Simplify the numerator.

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Step 2.9.3.1

Multiply $- 1$ by $5$ .

$\frac{5 y \sec^{2} (\frac{x}{y}) - 5 x y' \sec^{2} (\frac{x}{y})}{y^{2}}$

Step 2.9.3.2

Reorder factors in $5 y \sec^{2} (\frac{x}{y}) - 5 x y' \sec^{2} (\frac{x}{y})$ .

$\frac{5 y \sec^{2} (\frac{x}{y}) - 5 x \sec^{2} (\frac{x}{y}) y'}{y^{2}}$

Step 2.9.4

Factor $5 \sec^{2} (\frac{x}{y})$ out of $5 y \sec^{2} (\frac{x}{y}) - 5 x \sec^{2} (\frac{x}{y}) y'$ .

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Step 2.9.4.1

Factor $5 \sec^{2} (\frac{x}{y})$ out of $5 y \sec^{2} (\frac{x}{y})$ .

$\frac{5 \sec^{2} (\frac{x}{y}) (y) - 5 x \sec^{2} (\frac{x}{y}) y'}{y^{2}}$

Step 2.9.4.2

Factor $5 \sec^{2} (\frac{x}{y})$ out of $- 5 x \sec^{2} (\frac{x}{y}) y'$ .

$\frac{5 \sec^{2} (\frac{x}{y}) (y) + 5 \sec^{2} (\frac{x}{y}) (- x y')}{y^{2}}$

Step 2.9.4.3

Factor $5 \sec^{2} (\frac{x}{y})$ out of $5 \sec^{2} (\frac{x}{y}) (y) + 5 \sec^{2} (\frac{x}{y}) (- x y')$ .

$\frac{5 \sec^{2} (\frac{x}{y}) (y - x y')}{y^{2}}$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x$ with respect to $x$ is $12 \frac{d}{d x} [x]$ .

$12 \frac{d}{d x} [x]$

Step 3.2

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

$12 \cdot 1$

Step 3.3

Multiply $12$ by $1$ .

$12$

Step 4

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

$\frac{5 \sec^{2} (\frac{x}{y}) (y - x y')}{y^{2}} = 12$

Step 5

Solve for $y'$ .

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Step 5.1

Multiply both sides by $y^{2}$ .

$\frac{5 \sec^{2} (\frac{x}{y}) (y - x y')}{y^{2}} y^{2} = 12 y^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify $\frac{5 \sec^{2} (\frac{x}{y}) (y - x y')}{y^{2}} y^{2}$ .

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Step 5.2.1.1

Cancel the common factor of $y^{2}$ .

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Step 5.2.1.1.1

Cancel the common factor.

$\frac{5 \sec^{2} (\frac{x}{y}) (y - x y')}{y^{2}} y^{2} = 12 y^{2}$

Step 5.2.1.1.2

Rewrite the expression.

$5 \sec^{2} (\frac{x}{y}) (y - x y') = 12 y^{2}$

Step 5.2.1.2

Apply the distributive property.

$5 \sec^{2} (\frac{x}{y}) y + 5 \sec^{2} (\frac{x}{y}) (- x y') = 12 y^{2}$

Step 5.2.1.3

Simplify the expression.

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Step 5.2.1.3.1

Multiply $- 1$ by $5$ .

$5 \sec^{2} (\frac{x}{y}) y - 5 \sec^{2} (\frac{x}{y}) (x y') = 12 y^{2}$

Step 5.2.1.3.2

Reorder factors in $5 \sec^{2} (\frac{x}{y}) y - 5 \sec^{2} (\frac{x}{y}) (x y')$ .

$5 y \sec^{2} (\frac{x}{y}) - 5 x y' \sec^{2} (\frac{x}{y}) = 12 y^{2}$

Step 5.2.1.3.3

Move $x$ .

$5 y \sec^{2} (\frac{x}{y}) - 5 y' x \sec^{2} (\frac{x}{y}) = 12 y^{2}$

Step 5.2.1.3.4

Reorder $5 y \sec^{2} (\frac{x}{y})$ and $- 5 y' x \sec^{2} (\frac{x}{y})$ .

$- 5 y' x \sec^{2} (\frac{x}{y}) + 5 y \sec^{2} (\frac{x}{y}) = 12 y^{2}$

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Subtract $5 y \sec^{2} (\frac{x}{y})$ from both sides of the equation.

$- 5 y' x \sec^{2} (\frac{x}{y}) = 12 y^{2} - 5 y \sec^{2} (\frac{x}{y})$

Step 5.3.2

Divide each term in $- 5 y' x \sec^{2} (\frac{x}{y}) = 12 y^{2} - 5 y \sec^{2} (\frac{x}{y})$ by $- 5 x \sec^{2} (\frac{x}{y})$ and simplify.

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Step 5.3.2.1

Divide each term in $- 5 y' x \sec^{2} (\frac{x}{y}) = 12 y^{2} - 5 y \sec^{2} (\frac{x}{y})$ by $- 5 x \sec^{2} (\frac{x}{y})$ .

$\frac{- 5 y' x \sec^{2} (\frac{x}{y})}{- 5 x \sec^{2} (\frac{x}{y})} = \frac{12 y^{2}}{- 5 x \sec^{2} (\frac{x}{y})} + \frac{- 5 y \sec^{2} (\frac{x}{y})}{- 5 x \sec^{2} (\frac{x}{y})}$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Cancel the common factor of $- 5$ .

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Step 5.3.2.2.1.1

Cancel the common factor.

$\frac{- 5 y' x \sec^{2} (\frac{x}{y})}{- 5 x \sec^{2} (\frac{x}{y})} = \frac{12 y^{2}}{- 5 x \sec^{2} (\frac{x}{y})} + \frac{- 5 y \sec^{2} (\frac{x}{y})}{- 5 x \sec^{2} (\frac{x}{y})}$

Step 5.3.2.2.1.2

Rewrite the expression.

$\frac{y' x \sec^{2} (\frac{x}{y})}{x \sec^{2} (\frac{x}{y})} = \frac{12 y^{2}}{- 5 x \sec^{2} (\frac{x}{y})} + \frac{- 5 y \sec^{2} (\frac{x}{y})}{- 5 x \sec^{2} (\frac{x}{y})}$

Step 5.3.2.2.2

Cancel the common factor of $x$ .

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Step 5.3.2.2.2.1

Cancel the common factor.

$\frac{y' x \sec^{2} (\frac{x}{y})}{x \sec^{2} (\frac{x}{y})} = \frac{12 y^{2}}{- 5 x \sec^{2} (\frac{x}{y})} + \frac{- 5 y \sec^{2} (\frac{x}{y})}{- 5 x \sec^{2} (\frac{x}{y})}$

Step 5.3.2.2.2.2

Rewrite the expression.

$\frac{y' \sec^{2} (\frac{x}{y})}{\sec^{2} (\frac{x}{y})} = \frac{12 y^{2}}{- 5 x \sec^{2} (\frac{x}{y})} + \frac{- 5 y \sec^{2} (\frac{x}{y})}{- 5 x \sec^{2} (\frac{x}{y})}$

Step 5.3.2.2.3

Cancel the common factor of $\sec^{2} (\frac{x}{y})$ .

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Step 5.3.2.2.3.1

Cancel the common factor.

$\frac{y' \sec^{2} (\frac{x}{y})}{\sec^{2} (\frac{x}{y})} = \frac{12 y^{2}}{- 5 x \sec^{2} (\frac{x}{y})} + \frac{- 5 y \sec^{2} (\frac{x}{y})}{- 5 x \sec^{2} (\frac{x}{y})}$

Step 5.3.2.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{12 y^{2}}{- 5 x \sec^{2} (\frac{x}{y})} + \frac{- 5 y \sec^{2} (\frac{x}{y})}{- 5 x \sec^{2} (\frac{x}{y})}$

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Simplify each term.

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Step 5.3.2.3.1.1

Move the negative in front of the fraction.

$y' = - \frac{12 y^{2}}{5 x \sec^{2} (\frac{x}{y})} + \frac{- 5 y \sec^{2} (\frac{x}{y})}{- 5 x \sec^{2} (\frac{x}{y})}$

Step 5.3.2.3.1.2

Cancel the common factor of $- 5$ .

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Step 5.3.2.3.1.2.1

Cancel the common factor.

$y' = - \frac{12 y^{2}}{5 x \sec^{2} (\frac{x}{y})} + \frac{- 5 y \sec^{2} (\frac{x}{y})}{- 5 x \sec^{2} (\frac{x}{y})}$

Step 5.3.2.3.1.2.2

Rewrite the expression.

$y' = - \frac{12 y^{2}}{5 x \sec^{2} (\frac{x}{y})} + \frac{y \sec^{2} (\frac{x}{y})}{x \sec^{2} (\frac{x}{y})}$

Step 5.3.2.3.1.3

Cancel the common factor of $\sec^{2} (\frac{x}{y})$ .

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Step 5.3.2.3.1.3.1

Cancel the common factor.

$y' = - \frac{12 y^{2}}{5 x \sec^{2} (\frac{x}{y})} + \frac{y \sec^{2} (\frac{x}{y})}{x \sec^{2} (\frac{x}{y})}$

Step 5.3.2.3.1.3.2

Rewrite the expression.

$y' = - \frac{12 y^{2}}{5 x \sec^{2} (\frac{x}{y})} + \frac{y}{x}$

Step 6

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

$\frac{d y}{d x} = - \frac{12 y^{2}}{5 x \sec^{2} (\frac{x}{y})} + \frac{y}{x}$