Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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 $g (x) = y$ .

Tap for more steps...

Step 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.2.3

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

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

Step 2.3.3

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

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

Step 2.4

Reorder terms.

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

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Differentiate.

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

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

$1 - y'$

Step 4

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

$- e^{y} y' + e^{x} = 1 - y'$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

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

$- y' e^{y} + e^{x} = 1 - y'$

Step 5.2

Add $y'$ to both sides of the equation.

$- y' e^{y} + e^{x} + y' = 1$

Step 5.3

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

$- y' e^{y} + y' = 1 - e^{x}$

Step 5.4

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

Tap for more steps...

Step 5.4.1

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

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

Step 5.4.2

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

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

Step 5.4.3

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

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

Step 5.4.4

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

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

Step 5.5

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

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

Step 5.6

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

Tap for more steps...

Step 5.6.1

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

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

Step 5.6.2

Simplify the left side.

Tap for more steps...

Step 5.6.2.1

Cancel the common factor of $- e^{y} + 1$ .

Tap for more steps...

Step 5.6.2.1.1

Cancel the common factor.

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

Step 5.6.2.1.2

Divide $y'$ by $1$ .

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

Step 5.6.3

Simplify the right side.

Tap for more steps...

Step 5.6.3.1

Move the negative in front of the fraction.

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

Step 5.6.3.2

Combine the numerators over the common denominator.

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

Step 5.6.3.3

Factor $- 1$ out of $- e^{y}$ .

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

Step 5.6.3.4

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 5.6.3.5

Factor $- 1$ out of $- (e^{y}) - 1 (- 1)$ .

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

Step 5.6.3.6

Rewrite negatives.

Tap for more steps...

Step 5.6.3.6.1

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

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

Step 5.6.3.6.2

Move the negative in front of the fraction.

$y' = - \frac{1 - e^{x}}{e^{y} - 1}$

Step 6

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

$\frac{d y}{d x} = - \frac{1 - e^{x}}{e^{y} - 1}$