Find dx/dy xy+8e^y=8e

$x y + 8 e^{y} = 8 e$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (x y + 8 e^{y}) = \frac{d}{d y} (8 e)$

Step 2

Differentiate the left side of the equation.

By the Sum Rule, the derivative of $x y + 8 e^{y}$ with respect to $y$ is $\frac{d}{d y} [x y] + \frac{d}{d y} [8 e^{y}]$ .

$\frac{d}{d y} [x y] + \frac{d}{d y} [8 e^{y}]$

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

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 $g (y) = y$ .

$x \frac{d}{d y} [y] + y \frac{d}{d y} [x] + \frac{d}{d y} [8 e^{y}]$

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

$x \cdot 1 + y \frac{d}{d y} [x] + \frac{d}{d y} [8 e^{y}]$

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

$x \cdot 1 + y x' + \frac{d}{d y} [8 e^{y}]$

Multiply $x$ by $1$ .

$x + y x' + \frac{d}{d y} [8 e^{y}]$

Evaluate $\frac{d}{d y} [8 e^{y}]$ .

Since $8$ is constant with respect to $y$ , the derivative of $8 e^{y}$ with respect to $y$ is $8 \frac{d}{d y} [e^{y}]$ .

$x + y x' + 8 \frac{d}{d y} [e^{y}]$

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

$x + y x' + 8 e^{y}$

Reorder terms.

$y x' + x + 8 e^{y}$

Step 3

Since $8 e$ is constant with respect to $y$ , the derivative of $8 e$ with respect to $y$ is $0$ .

$0$

Step 4

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

$y x' + x + 8 e^{y} = 0$

Step 5

Solve for $x'$ .

Move all terms not containing $x'$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$y x' + 8 e^{y} = - x$

Subtract $8 e^{y}$ from both sides of the equation.

$y x' = - x - 8 e^{y}$

Divide each term in $y x' = - x - 8 e^{y}$ by $y$ and simplify.

Divide each term in $y x' = - x - 8 e^{y}$ by $y$ .

$\frac{y x'}{y} = \frac{- x}{y} + \frac{- 8 e^{y}}{y}$

Simplify the left side.

Cancel the common factor of $y$ .

Cancel the common factor.

$\frac{y x'}{y} = \frac{- x}{y} + \frac{- 8 e^{y}}{y}$

Divide $x'$ by $1$ .

$x' = \frac{- x}{y} + \frac{- 8 e^{y}}{y}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x' = - \frac{x}{y} + \frac{- 8 e^{y}}{y}$

Move the negative in front of the fraction.

$x' = - \frac{x}{y} - \frac{8 e^{y}}{y}$

Step 6

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

$\frac{d x}{d y} = - \frac{x}{y} - \frac{8 e^{y}}{y}$

Find dx/dy xy+8e^y=8e

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