Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

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

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

Step 2.5

Reorder terms.

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.2

Evaluate $\frac{d}{d x} [x y]$ .

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Step 3.2.1

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 $g (x) = y$ .

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

Step 3.2.2

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

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

Step 3.2.3

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

$1 + x y' + y \cdot 1$

Step 3.2.4

Multiply $y$ by $1$ .

$1 + x y' + y$

Step 3.3

Reorder terms.

$x y' + y + 1$

Step 4

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

$e^{y} \cos (x) + e^{y} \sin (x) y' = x y' + y + 1$

Step 5

Solve for $y'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

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

$e^{y} \cos (x) + y' e^{y} \sin (x) = x y' + y + 1$

Step 5.2

Subtract $x y'$ from both sides of the equation.

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

Step 5.3

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

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

Step 5.4

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

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Step 5.4.1

Factor $y'$ out of $y' e^{y} \sin (x)$ .

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

Step 5.4.2

Factor $y'$ out of $- x y'$ .

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

Step 5.4.3

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

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

Step 5.5

Divide each term in $y' (e^{y} \sin (x) - x) = y + 1 - e^{y} \cos (x)$ by $e^{y} \sin (x) - x$ and simplify.

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Step 5.5.1

Divide each term in $y' (e^{y} \sin (x) - x) = y + 1 - e^{y} \cos (x)$ by $e^{y} \sin (x) - x$ .

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

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

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

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Step 5.5.2.1.1

Cancel the common factor.

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

Step 5.5.2.1.2

Divide $y'$ by $1$ .

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

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Move the negative in front of the fraction.

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

Step 5.5.3.2

Combine the numerators over the common denominator.

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

Step 5.5.3.3

Combine the numerators over the common denominator.

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

Step 6

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

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