Algebra Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = e^{y}$ and $g (y) = \sin (x)$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Reorder terms.

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x$ and $g (y) = y$ .

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Multiply $x$ by $1$ .

$x' + x + y x'$

Step 3.4

Reorder terms.

$y x' + x + x'$

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

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

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

Step 5.2

Since $x'$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 5.3

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

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

Step 5.4

Subtract $x$ from both sides of the equation.

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

Raise $x'$ to the power of $1$ .

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

Step 5.5.3

Factor $x'$ out of ${x'}^{1}$ .

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

Step 5.5.4

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

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

Step 5.5.5

Factor $x'$ out of $x' y + x' \cdot 1$ .

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

Step 5.5.6

Factor $x'$ out of $x' (y + 1) + x' (- 1 e^{y} \cos (x))$ .

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

Step 5.6

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

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

Step 5.7

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

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Step 5.7.1

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

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

Step 5.7.2

Simplify the left side.

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Step 5.7.2.1

Cancel the common factor of $y + 1 - e^{y} \cos (x)$ .

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Step 5.7.2.1.1

Cancel the common factor.

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

Step 5.7.2.1.2

Divide $x'$ by $1$ .

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

Step 5.7.3

Simplify the right side.

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Step 5.7.3.1

Move the negative in front of the fraction.

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

Step 5.7.3.2

Combine the numerators over the common denominator.

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

Step 6

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

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