Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

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

$\frac{d}{d x} [x] (y x^{y - 1}) + \frac{d}{d x} [y] (x^{y} \ln (x))$

Step 2.2

Differentiate using the Power Rule.

Tap for more steps...

Step 2.2.1

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

$1 (y x^{y - 1}) + \frac{d}{d x} [y] (x^{y} \ln (x))$

Step 2.2.2

Multiply $y$ by $1$ .

$y x^{y - 1} + \frac{d}{d x} [y] (x^{y} \ln (x))$

Step 2.3

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

$y x^{y - 1} + y' (x^{y} \ln (x))$

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Reorder terms.

$x^{y - 1} y + x^{y} \ln (x) y'$

Step 2.4.2

Reorder factors in $x^{y - 1} y + x^{y} \ln (x) y'$ .

$y x^{y - 1} + x^{y} \ln (x) y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

Differentiate.

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

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

$e^{x - y} (1 + \frac{d}{d x} [- y])$

Step 3.2.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- y$ with respect to $x$ is $- \frac{d}{d x} [y]$ .

$e^{x - y} (1 - \frac{d}{d x} [y])$

Step 3.3

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

$e^{x - y} (1 - y')$

Step 4

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

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

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Reorder factors in $y x^{y - 1} + x^{y} \ln (x) y'$ .

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

Step 5.2

Simplify $e^{x - y} (1 - y')$ .

Tap for more steps...

Step 5.2.1

Apply the distributive property.

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

Step 5.2.2

Simplify the expression.

Tap for more steps...

Step 5.2.2.1

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

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

Step 5.2.2.2

Rewrite using the commutative property of multiplication.

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

Step 5.2.2.3

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

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

Step 5.3

Add $y' e^{x - y}$ to both sides of the equation.

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

Step 5.4

Subtract $y x^{y - 1}$ from both sides of the equation.

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

Step 5.5

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

Tap for more steps...

Step 5.5.1

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

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

Step 5.5.2

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

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

Step 5.5.3

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

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

Step 5.6

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

Tap for more steps...

Step 5.6.1

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

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

Step 5.6.2

Simplify the left side.

Tap for more steps...

Step 5.6.2.1

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

Tap for more steps...

Step 5.6.2.1.1

Cancel the common factor.

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

Step 5.6.2.1.2

Divide $y'$ by $1$ .

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

Step 5.6.3

Simplify the right side.

Tap for more steps...

Step 5.6.3.1

Combine the numerators over the common denominator.

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

Step 6

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

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