Calculus Examples

$\tan (\frac{x}{y}) = x + y$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

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

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

Step 2.3

Differentiate using the Power Rule.

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

Multiply $y$ by $1$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

$1 + y'$

Step 4

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

$\frac{\sec^{2} (\frac{x}{y}) (y - x y')}{y^{2}} = 1 + y'$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

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

$\frac{\sec^{2} (\frac{x}{y}) (y - x y')}{y^{2}} y^{2} = (1 + y') y^{2}$

Step 5.2

Simplify.

Tap for more steps...

Step 5.2.1

Simplify the left side.

Tap for more steps...

Step 5.2.1.1

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

Tap for more steps...

Step 5.2.1.1.1

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

Tap for more steps...

Step 5.2.1.1.1.1

Cancel the common factor.

$\frac{\sec^{2} (\frac{x}{y}) (y - x y')}{y^{2}} y^{2} = (1 + y') y^{2}$

Step 5.2.1.1.1.2

Rewrite the expression.

$\sec^{2} (\frac{x}{y}) (y - x y') = (1 + y') y^{2}$

Step 5.2.1.1.2

Apply the distributive property.

$\sec^{2} (\frac{x}{y}) y + \sec^{2} (\frac{x}{y}) (- x y') = (1 + y') y^{2}$

Step 5.2.1.1.3

Simplify the expression.

Tap for more steps...

Step 5.2.1.1.3.1

Rewrite using the commutative property of multiplication.

$\sec^{2} (\frac{x}{y}) y - \sec^{2} (\frac{x}{y}) (x y') = (1 + y') y^{2}$

Step 5.2.1.1.3.2

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

$y \sec^{2} (\frac{x}{y}) - x y' \sec^{2} (\frac{x}{y}) = (1 + y') y^{2}$

Step 5.2.1.1.3.3

Move $x$ .

$y \sec^{2} (\frac{x}{y}) - y' x \sec^{2} (\frac{x}{y}) = (1 + y') y^{2}$

Step 5.2.1.1.3.4

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

$- y' x \sec^{2} (\frac{x}{y}) + y \sec^{2} (\frac{x}{y}) = (1 + y') y^{2}$

Step 5.2.2

Simplify the right side.

Tap for more steps...

Step 5.2.2.1

Simplify $(1 + y') y^{2}$ .

Tap for more steps...

Step 5.2.2.1.1

Apply the distributive property.

$- y' x \sec^{2} (\frac{x}{y}) + y \sec^{2} (\frac{x}{y}) = 1 y^{2} + y' y^{2}$

Step 5.2.2.1.2

Simplify the expression.

Tap for more steps...

Step 5.2.2.1.2.1

Multiply $y^{2}$ by $1$ .

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

Step 5.2.2.1.2.2

Reorder $y^{2}$ and $y' y^{2}$ .

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

Step 5.3

Solve for $y'$ .

Tap for more steps...

Step 5.3.1

Subtract $y' y^{2}$ from both sides of the equation.

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

Step 5.3.2

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

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

Step 5.3.3

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

Tap for more steps...

Step 5.3.3.1

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

$y' (- 1 x \sec^{2} (\frac{x}{y})) - y' y^{2} = y^{2} - y \sec^{2} (\frac{x}{y})$

Step 5.3.3.2

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

$y' (- 1 x \sec^{2} (\frac{x}{y})) + y' (- 1 y^{2}) = y^{2} - y \sec^{2} (\frac{x}{y})$

Step 5.3.3.3

Factor $y'$ out of $y' (- 1 x \sec^{2} (\frac{x}{y})) + y' (- 1 y^{2})$ .

$y' (- 1 x \sec^{2} (\frac{x}{y}) - 1 y^{2}) = y^{2} - y \sec^{2} (\frac{x}{y})$

Step 5.3.4

Rewrite $- 1 x$ as $- x$ .

$y' (- x \sec^{2} (\frac{x}{y}) - 1 y^{2}) = y^{2} - y \sec^{2} (\frac{x}{y})$

Step 5.3.5

Rewrite $- 1 y^{2}$ as $- y^{2}$ .

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

Step 5.3.6

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

Tap for more steps...

Step 5.3.6.1

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

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

Step 5.3.6.2

Simplify the left side.

Tap for more steps...

Step 5.3.6.2.1

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

Tap for more steps...

Step 5.3.6.2.1.1

Cancel the common factor.

Step 5.3.6.2.1.2

Divide $y'$ by $1$ .

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

Step 5.3.6.3

Simplify the right side.

Tap for more steps...

Step 5.3.6.3.1

Combine into one fraction.

Tap for more steps...

Step 5.3.6.3.1.1

Move the negative in front of the fraction.

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

Step 5.3.6.3.1.2

Combine the numerators over the common denominator.

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

Step 5.3.6.3.2

Simplify the numerator.

Tap for more steps...

Step 5.3.6.3.2.1

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

Tap for more steps...

Step 5.3.6.3.2.1.1

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

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

Step 5.3.6.3.2.1.2

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

$y' = \frac{y \cdot y + y (- 1 \sec^{2} (\frac{x}{y}))}{- x \sec^{2} (\frac{x}{y}) - y^{2}}$

Step 5.3.6.3.2.1.3

Factor $y$ out of $y \cdot y + y (- 1 \sec^{2} (\frac{x}{y}))$ .

$y' = \frac{y (y - 1 \sec^{2} (\frac{x}{y}))}{- x \sec^{2} (\frac{x}{y}) - y^{2}}$

Step 5.3.6.3.2.2

Rewrite $- 1 \sec^{2} (\frac{x}{y})$ as $- \sec^{2} (\frac{x}{y})$ .

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

Step 5.3.6.3.3

Simplify with factoring out.

Tap for more steps...

Step 5.3.6.3.3.1

Factor $- 1$ out of $- x \sec^{2} (\frac{x}{y})$ .

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

Step 5.3.6.3.3.2

Factor $- 1$ out of $- y^{2}$ .

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

Step 5.3.6.3.3.3

Factor $- 1$ out of $- (x \sec^{2} (\frac{x}{y})) - (y^{2})$ .

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

Step 5.3.6.3.3.4

Rewrite negatives.

Tap for more steps...

Step 5.3.6.3.3.4.1

Rewrite $- (x \sec^{2} (\frac{x}{y}) + y^{2})$ as $- 1 (x \sec^{2} (\frac{x}{y}) + y^{2})$ .

$y' = \frac{y (y - \sec^{2} (\frac{x}{y}))}{- 1 (x \sec^{2} (\frac{x}{y}) + y^{2})}$

Step 5.3.6.3.3.4.2

Move the negative in front of the fraction.

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

Step 6

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

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