Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $x^{2} + y^{2}$ .

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

Step 2.1.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

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

Step 2.1.3

Replace all occurrences of $u_{1}$ with $x^{2} + y^{2}$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $x^{2} + y^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}]$ .

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

Step 2.2.2

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

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

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 2.3.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

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

Step 2.3.2

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

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

Step 2.3.3

Replace all occurrences of $u_{2}$ with $y$ .

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

Step 2.4

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

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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

Step 3.4

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

$x e^{y} y' + e^{y} \cdot 1$

Step 3.5

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

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

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $\cos (x^{2} + y^{2}) (2 x + 2 y y')$ .

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Step 5.1.1.1

Rewrite.

$0 + 0 + \cos (x^{2} + y^{2}) (2 x + 2 y y') = x e^{y} y' + e^{y}$

Step 5.1.1.2

Simplify by adding zeros.

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

Step 5.1.1.3

Apply the distributive property.

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

Step 5.1.1.4

Reorder.

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Step 5.1.1.4.1

Rewrite using the commutative property of multiplication.

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

Step 5.1.1.4.2

Rewrite using the commutative property of multiplication.

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

Step 5.1.1.4.3

Reorder factors in $2 \cos (x^{2} + y^{2}) x + 2 \cos (x^{2} + y^{2}) (y y')$ .

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

Step 5.2

Simplify the right side.

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Step 5.2.1

Reorder factors in $x e^{y} y' + e^{y}$ .

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

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

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

Step 5.5.3

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

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

Step 5.6

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

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Step 5.6.1

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

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

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Cancel the common factor of $2 y \cos (x^{2} + y^{2}) - x e^{y}$ .

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Step 5.6.2.1.1

Cancel the common factor.

Step 5.6.2.1.2

Divide $y'$ by $1$ .

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

Step 5.6.3

Simplify the right side.

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Step 5.6.3.1

Move the negative in front of the fraction.

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

Step 5.6.3.2

Combine the numerators over the common denominator.

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

Step 6

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

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