Calculus Examples

$e^{y} = \frac{x^{2}}{y}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (e^{y}) = \frac{d}{d x} (\frac{x^{2}}{y})$

Step 2

Differentiate the left side of the equation.

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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 $f (x) = e^{x}$ and $g (x) = y$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $y$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [y]$

Step 2.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$e^{u} \frac{d}{d x} [y]$

Step 2.1.3

Replace all occurrences of $u$ with $y$ .

$e^{y} \frac{d}{d x} [y]$

Step 2.2

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

$e^{y} y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = x^{2}$ and $g (x) = y$ .

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

Step 3.2

Differentiate using the Power Rule.

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Step 3.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 3.2.2

Move $2$ to the left of $y$ .

$\frac{2 \cdot y x - x^{2} \frac{d}{d x} [y]}{y^{2}}$

Step 3.3

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

$\frac{2 y x - x^{2} y'}{y^{2}}$

Step 3.4

Simplify.

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Step 3.4.1

Reorder terms.

$\frac{2 x y - x^{2} y'}{y^{2}}$

Step 3.4.2

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

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Step 3.4.2.1

Factor $x$ out of $2 x y$ .

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

Step 3.4.2.2

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

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

Step 3.4.2.3

Factor $x$ out of $x (2 y) + x (- x y')$ .

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

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Reorder factors in $e^{y} y' y^{2}$ .

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

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

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Step 5.2.2.1.1

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

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Step 5.2.2.1.1.1

Cancel the common factor.

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

Step 5.2.2.1.1.2

Rewrite the expression.

$y' y^{2} e^{y} = x (2 y - x y')$

Step 5.2.2.1.2

Apply the distributive property.

$y' y^{2} e^{y} = x (2 y) + x (- x y')$

Step 5.2.2.1.3

Reorder.

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Step 5.2.2.1.3.1

Rewrite using the commutative property of multiplication.

$y' y^{2} e^{y} = 2 x y + x (- x y')$

Step 5.2.2.1.3.2

Rewrite using the commutative property of multiplication.

$y' y^{2} e^{y} = 2 x y - x (x y')$

Step 5.2.2.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 5.2.2.1.4.1

Move $x$ .

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

Step 5.2.2.1.4.2

Multiply $x$ by $x$ .

$y' y^{2} e^{y} = 2 x y - x^{2} y'$

Step 5.2.2.1.5

Move $x^{2}$ .

$y' y^{2} e^{y} = 2 x y - y' x^{2}$

Step 5.2.2.1.6

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

$y' y^{2} e^{y} = - y' x^{2} + 2 x y$

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Add $y' x^{2}$ to both sides of the equation.

$y' y^{2} e^{y} + y' x^{2} = 2 x y$

Step 5.3.2

Factor $y'$ out of $y' y^{2} e^{y} + y' x^{2}$ .

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Step 5.3.2.1

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

$y' (y^{2} e^{y}) + y' (x^{2}) = 2 x y$

Step 5.3.2.2

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

$y' (y^{2} e^{y}) + y' (x^{2}) = 2 x y$

Step 5.3.2.3

Factor $y'$ out of $y' (y^{2} e^{y}) + y' (x^{2})$ .

$y' (y^{2} e^{y} + x^{2}) = 2 x y$

Step 5.3.3

Divide each term in $y' (y^{2} e^{y} + x^{2}) = 2 x y$ by $y^{2} e^{y} + x^{2}$ and simplify.

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Step 5.3.3.1

Divide each term in $y' (y^{2} e^{y} + x^{2}) = 2 x y$ by $y^{2} e^{y} + x^{2}$ .

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

Step 5.3.3.2

Simplify the left side.

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Step 5.3.3.2.1

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

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Step 5.3.3.2.1.1

Cancel the common factor.

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

Step 5.3.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{2 x y}{y^{2} e^{y} + x^{2}}$

Step 6

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

$\frac{d y}{d x} = \frac{2 x y}{y^{2} e^{y} + x^{2}}$