Calculus Examples

$e^{x y} = e^{4 x} - e^{5 y}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (e^{x y}) = \frac{d}{d x} (e^{4 x} - e^{5 y})$

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

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 2.1.1

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

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

Step 2.1.2

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 2.1.3

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

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

Step 2.2

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 $g (x) = y$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Multiply $y$ by $1$ .

$e^{x y} (x y' + y)$

Step 2.6

Simplify.

Tap for more steps...

Step 2.6.1

Apply the distributive property.

$e^{x y} (x y') + e^{x y} y$

Step 2.6.2

Reorder terms.

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

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $e^{4 x} - e^{5 y}$ with respect to $x$ is $\frac{d}{d x} [e^{4 x}] + \frac{d}{d x} [- e^{5 y}]$ .

$\frac{d}{d x} [e^{4 x}] + \frac{d}{d x} [- e^{5 y}]$

Step 3.2

Evaluate $\frac{d}{d x} [e^{4 x}]$ .

Tap for more steps...

Step 3.2.1

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 $g (x) = 4 x$ .

Tap for more steps...

Step 3.2.1.1

To apply the Chain Rule, set $u_{1}$ as $4 x$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [4 x] + \frac{d}{d x} [- e^{5 y}]$

Step 3.2.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [4 x] + \frac{d}{d x} [- e^{5 y}]$

Step 3.2.1.3

Replace all occurrences of $u_{1}$ with $4 x$ .

$e^{4 x} \frac{d}{d x} [4 x] + \frac{d}{d x} [- e^{5 y}]$

Step 3.2.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$e^{4 x} (4 \frac{d}{d x} [x]) + \frac{d}{d x} [- e^{5 y}]$

Step 3.2.3

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

$e^{4 x} (4 \cdot 1) + \frac{d}{d x} [- e^{5 y}]$

Step 3.2.4

Multiply $4$ by $1$ .

$e^{4 x} \cdot 4 + \frac{d}{d x} [- e^{5 y}]$

Step 3.2.5

Move $4$ to the left of $e^{4 x}$ .

$4 e^{4 x} + \frac{d}{d x} [- e^{5 y}]$

Step 3.3

Evaluate $\frac{d}{d x} [- e^{5 y}]$ .

Tap for more steps...

Step 3.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- e^{5 y}$ with respect to $x$ is $- \frac{d}{d x} [e^{5 y}]$ .

$4 e^{4 x} - \frac{d}{d x} [e^{5 y}]$

Step 3.3.2

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 $g (x) = 5 y$ .

Tap for more steps...

Step 3.3.2.1

To apply the Chain Rule, set $u_{2}$ as $5 y$ .

$4 e^{4 x} - (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [5 y])$

Step 3.3.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$4 e^{4 x} - (e^{u_{2}} \frac{d}{d x} [5 y])$

Step 3.3.2.3

Replace all occurrences of $u_{2}$ with $5 y$ .

$4 e^{4 x} - (e^{5 y} \frac{d}{d x} [5 y])$

Step 3.3.3

Since $5$ is constant with respect to $x$ , the derivative of $5 y$ with respect to $x$ is $5 \frac{d}{d x} [y]$ .

$4 e^{4 x} - (e^{5 y} (5 \frac{d}{d x} [y]))$

Step 3.3.4

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

$4 e^{4 x} - (e^{5 y} (5 y'))$

Step 3.3.5

Multiply $5$ by $- 1$ .

$4 e^{4 x} - 5 e^{5 y} y'$

Step 3.4

Reorder terms.

$- 5 e^{5 y} y' + 4 e^{4 x}$

Step 4

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

$e^{x y} x y' + e^{x y} y = - 5 e^{5 y} y' + 4 e^{4 x}$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

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

$x y' e^{x y} + y e^{x y} = - 5 e^{5 y} y' + 4 e^{4 x}$

Step 5.2

Reorder factors in $- 5 e^{5 y} y' + 4 e^{4 x}$ .

$x y' e^{x y} + y e^{x y} = - 5 y' e^{5 y} + 4 e^{4 x}$

Step 5.3

Add $5 y' e^{5 y}$ to both sides of the equation.

$x y' e^{x y} + y e^{x y} + 5 y' e^{5 y} = 4 e^{4 x}$

Step 5.4

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

$x y' e^{x y} + 5 y' e^{5 y} = 4 e^{4 x} - y e^{x y}$

Step 5.5

Factor $y'$ out of $x y' e^{x y} + 5 y' e^{5 y}$ .

Tap for more steps...

Step 5.5.1

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

$y' (x e^{x y}) + 5 y' e^{5 y} = 4 e^{4 x} - y e^{x y}$

Step 5.5.2

Factor $y'$ out of $5 y' e^{5 y}$ .

$y' (x e^{x y}) + y' (5 e^{5 y}) = 4 e^{4 x} - y e^{x y}$

Step 5.5.3

Factor $y'$ out of $y' (x e^{x y}) + y' (5 e^{5 y})$ .

$y' (x e^{x y} + 5 e^{5 y}) = 4 e^{4 x} - y e^{x y}$

Step 5.6

Divide each term in $y' (x e^{x y} + 5 e^{5 y}) = 4 e^{4 x} - y e^{x y}$ by $x e^{x y} + 5 e^{5 y}$ and simplify.

Tap for more steps...

Step 5.6.1

Divide each term in $y' (x e^{x y} + 5 e^{5 y}) = 4 e^{4 x} - y e^{x y}$ by $x e^{x y} + 5 e^{5 y}$ .

$\frac{y' (x e^{x y} + 5 e^{5 y})}{x e^{x y} + 5 e^{5 y}} = \frac{4 e^{4 x}}{x e^{x y} + 5 e^{5 y}} + \frac{- y e^{x y}}{x e^{x y} + 5 e^{5 y}}$

Step 5.6.2

Simplify the left side.

Tap for more steps...

Step 5.6.2.1

Cancel the common factor of $x e^{x y} + 5 e^{5 y}$ .

Tap for more steps...

Step 5.6.2.1.1

Cancel the common factor.

$\frac{y' (x e^{x y} + 5 e^{5 y})}{x e^{x y} + 5 e^{5 y}} = \frac{4 e^{4 x}}{x e^{x y} + 5 e^{5 y}} + \frac{- y e^{x y}}{x e^{x y} + 5 e^{5 y}}$

Step 5.6.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{4 e^{4 x}}{x e^{x y} + 5 e^{5 y}} + \frac{- y e^{x y}}{x e^{x y} + 5 e^{5 y}}$

Step 5.6.3

Simplify the right side.

Tap for more steps...

Step 5.6.3.1

Combine the numerators over the common denominator.

$y' = \frac{4 e^{4 x} - y e^{x y}}{x e^{x y} + 5 e^{5 y}}$

Step 6

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

$\frac{d y}{d x} = \frac{4 e^{4 x} - y e^{x y}}{x e^{x y} + 5 e^{5 y}}$