Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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Step 2.1.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

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

Step 2.1.2

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

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

Step 2.1.3

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 2.2

Differentiate using the Power Rule.

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Step 2.2.1

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

$- \sin (x^{2}) (2 x)$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Multiply $2$ by $- 1$ .

$- 2 \sin (x^{2}) x$

Step 2.2.2.2

Reorder the factors of $- 2 \sin (x^{2}) x$ .

$- 2 x \sin (x^{2})$

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.

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

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $x e^{y} y' + e^{y} = - 2 x \sin (x^{2})$ .

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

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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

Step 5.3

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

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $x$ .

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Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Rewrite the expression.

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

Step 5.4.2.2

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

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Step 5.4.2.2.1

Cancel the common factor.

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

Step 5.4.2.2.2

Divide $y'$ by $1$ .

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

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Simplify each term.

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Step 5.4.3.1.1

Cancel the common factor of $x$ .

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Step 5.4.3.1.1.1

Cancel the common factor.

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

Step 5.4.3.1.1.2

Rewrite the expression.

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

Step 5.4.3.1.2

Move the negative in front of the fraction.

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

Step 5.4.3.1.3

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

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Step 5.4.3.1.3.1

Cancel the common factor.

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

Step 5.4.3.1.3.2

Rewrite the expression.

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

Step 5.4.3.1.4

Move the negative in front of the fraction.

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

Step 6

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

$\frac{d y}{d x} = - \frac{2 \sin (x^{2})}{e^{y}} - \frac{1}{x}$