Calculus Examples

$y = \ln (x + y)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\ln (x + y))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

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 $f (x) = \ln (x)$ and $g (x) = x + y$ .

Tap for more steps...

Step 3.1.1

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

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x + y]$

Step 3.1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

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

Step 3.1.3

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

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

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

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

Step 3.2.2

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

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

Step 3.3

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

$\frac{1}{x + y} (1 + y')$

Step 3.4

Simplify.

Tap for more steps...

Step 3.4.1

Reorder the factors of $\frac{1}{x + y} (1 + y')$ .

$(1 + y') \frac{1}{x + y}$

Step 3.4.2

Multiply $1 + y'$ by $\frac{1}{x + y}$ .

$\frac{1 + y'}{x + y}$

Step 4

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

$y' = \frac{1 + y'}{x + y}$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Multiply both sides by $x + y$ .

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

Step 5.2

Simplify.

Tap for more steps...

Step 5.2.1

Simplify the left side.

Tap for more steps...

Step 5.2.1.1

Apply the distributive property.

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

Step 5.2.2

Simplify the right side.

Tap for more steps...

Step 5.2.2.1

Simplify $\frac{1 + y'}{x + y} (x + y)$ .

Tap for more steps...

Step 5.2.2.1.1

Cancel the common factor of $x + y$ .

Tap for more steps...

Step 5.2.2.1.1.1

Cancel the common factor.

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

Step 5.2.2.1.1.2

Rewrite the expression.

$y' x + y' y = 1 + y'$

Step 5.2.2.1.2

Reorder $1$ and $y'$ .

$y' x + y' y = y' + 1$

Step 5.3

Solve for $y'$ .

Tap for more steps...

Step 5.3.1

Subtract $y'$ from both sides of the equation.

$y' x + y' y - y' = 1$

Step 5.3.2

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

Tap for more steps...

Step 5.3.2.1

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

$y' (x) + y' (y) - y' = 1$

Step 5.3.2.2

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

$y' (x) + y' (y) - y' = 1$

Step 5.3.2.3

Factor $y'$ out of $- y'$ .

$y' (x) + y' (y) + y' \cdot - 1 = 1$

Step 5.3.2.4

Factor $y'$ out of $y' (x) + y' (y)$ .

$y' (x + y) + y' \cdot - 1 = 1$

Step 5.3.2.5

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

$y' (x + y - 1) = 1$

Step 5.3.3

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

Tap for more steps...

Step 5.3.3.1

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

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

Step 5.3.3.2

Simplify the left side.

Tap for more steps...

Step 5.3.3.2.1

Cancel the common factor of $x + y - 1$ .

Tap for more steps...

Step 5.3.3.2.1.1

Cancel the common factor.

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

Step 5.3.3.2.1.2

Divide $y'$ by $1$ .

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

Step 6

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

$\frac{d y}{d x} = \frac{1}{x + y - 1}$