Calculus Examples

$\frac{4 y}{x + 2}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = 4 y$ and $g (y) = x + 2$ .

$\frac{(x + 2) \frac{d}{d y} [4 y] - 1 \cdot 4 y \frac{d}{d y} [x + 2]}{{(x + 2)}^{2}}$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

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

$\frac{(x + 2) (4 \frac{d}{d y} [y]) - 1 \cdot 4 y \frac{d}{d y} [x + 2]}{{(x + 2)}^{2}}$

Step 2.2

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

$\frac{(x + 2) (4 \cdot 1) - 1 \cdot 4 y \frac{d}{d y} [x + 2]}{{(x + 2)}^{2}}$

Step 2.3

Multiply $4$ by $1$ .

$\frac{(x + 2) \cdot 4 - 1 \cdot 4 y \frac{d}{d y} [x + 2]}{{(x + 2)}^{2}}$

Step 2.4

By the Sum Rule, the derivative of $x + 2$ with respect to $y$ is $\frac{d}{d y} [x] + \frac{d}{d y} [2]$ .

$\frac{(x + 2) \cdot 4 - 1 \cdot 4 y (\frac{d}{d y} [x] + \frac{d}{d y} [2])}{{(x + 2)}^{2}}$

Step 2.5

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

$\frac{(x + 2) \cdot 4 - 1 \cdot 4 y (0 + \frac{d}{d y} [2])}{{(x + 2)}^{2}}$

Step 2.6

Since $2$ is constant with respect to $y$ , the derivative of $2$ with respect to $y$ is $0$ .

$\frac{(x + 2) \cdot 4 - 1 \cdot 4 y (0 + 0)}{{(x + 2)}^{2}}$

Step 2.7

Add $0$ and $0$ .

$\frac{(x + 2) \cdot 4 - 1 \cdot 4 y \cdot 0}{{(x + 2)}^{2}}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Apply the distributive property.

$\frac{x \cdot 4 + 2 \cdot 4 - 1 \cdot 4 y \cdot 0}{{(x + 2)}^{2}}$

Step 3.2

Simplify the numerator.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Move $4$ to the left of $x$ .

$\frac{4 \cdot x + 2 \cdot 4 - 1 \cdot 4 y \cdot 0}{{(x + 2)}^{2}}$

Step 3.2.1.2

Multiply $2$ by $4$ .

$\frac{4 x + 8 - 1 \cdot 4 y \cdot 0}{{(x + 2)}^{2}}$

Step 3.2.1.3

Multiply $- 1$ by $4$ .

$\frac{4 x + 8 - 4 y \cdot 0}{{(x + 2)}^{2}}$

Step 3.2.1.4

Multiply $- 4 y \cdot 0$ .

Tap for more steps...

Step 3.2.1.4.1

Multiply $0$ by $- 4$ .

$\frac{4 x + 8 + 0 y}{{(x + 2)}^{2}}$

Step 3.2.1.4.2

Multiply $0$ by $y$ .

$\frac{4 x + 8 + 0}{{(x + 2)}^{2}}$

Step 3.2.2

Add $4 x + 8$ and $0$ .

$\frac{4 x + 8}{{(x + 2)}^{2}}$

Step 3.3

Factor $4$ out of $4 x + 8$ .

Tap for more steps...

Step 3.3.1

Factor $4$ out of $4 x$ .

$\frac{4 (x) + 8}{{(x + 2)}^{2}}$

Step 3.3.2

Factor $4$ out of $8$ .

$\frac{4 x + 4 \cdot 2}{{(x + 2)}^{2}}$

Step 3.3.3

Factor $4$ out of $4 x + 4 \cdot 2$ .

$\frac{4 (x + 2)}{{(x + 2)}^{2}}$

Step 3.4

Cancel the common factor of $x + 2$ and ${(x + 2)}^{2}$ .

Tap for more steps...

Step 3.4.1

Factor $x + 2$ out of $4 (x + 2)$ .

$\frac{(x + 2) \cdot 4}{{(x + 2)}^{2}}$

Step 3.4.2

Cancel the common factors.

Tap for more steps...

Step 3.4.2.1

Factor $x + 2$ out of ${(x + 2)}^{2}$ .

$\frac{(x + 2) \cdot 4}{(x + 2) (x + 2)}$

Step 3.4.2.2

Cancel the common factor.

$\frac{(x + 2) \cdot 4}{(x + 2) (x + 2)}$

Step 3.4.2.3

Rewrite the expression.

$\frac{4}{x + 2}$