Calculus Examples

$h (p) = \frac{p - 1}{p^{2} + 4}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d p} [\frac{f (p)}{g (p)}]$ is $\frac{g (p) \frac{d}{d p} [f (p)] - f (p) \frac{d}{d p} [g (p)]}{{g (p)}^{2}}$ where $f (p) = p - 1$ and $g (p) = p^{2} + 4$ .

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

Step 2

Differentiate.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $p - 1$ with respect to $p$ is $\frac{d}{d p} [p] + \frac{d}{d p} [- 1]$ .

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

Step 2.2

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

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

Step 2.3

Since $- 1$ is constant with respect to $p$ , the derivative of $- 1$ with respect to $p$ is $0$ .

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

Step 2.4

Simplify the expression.

Tap for more steps...

Step 2.4.1

Add $1$ and $0$ .

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

Step 2.4.2

Multiply $p^{2} + 4$ by $1$ .

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

Step 2.5

By the Sum Rule, the derivative of $p^{2} + 4$ with respect to $p$ is $\frac{d}{d p} [p^{2}] + \frac{d}{d p} [4]$ .

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

Step 2.6

Differentiate using the Power Rule which states that $\frac{d}{d p} [p^{n}]$ is $n p^{n - 1}$ where $n = 2$ .

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

Step 2.7

Since $4$ is constant with respect to $p$ , the derivative of $4$ with respect to $p$ is $0$ .

$\frac{p^{2} + 4 - (p - 1) (2 p + 0)}{{(p^{2} + 4)}^{2}}$

Step 2.8

Simplify the expression.

Tap for more steps...

Step 2.8.1

Add $2 p$ and $0$ .

$\frac{p^{2} + 4 - (p - 1) (2 p)}{{(p^{2} + 4)}^{2}}$

Step 2.8.2

Multiply $2$ by $- 1$ .

$\frac{p^{2} + 4 - 2 (p - 1) p}{{(p^{2} + 4)}^{2}}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Apply the distributive property.

$\frac{p^{2} + 4 + (- 2 p - 2 \cdot - 1) p}{{(p^{2} + 4)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{p^{2} + 4 - 2 p \cdot p - 2 \cdot - 1 p}{{(p^{2} + 4)}^{2}}$

Step 3.3

Simplify the numerator.

Tap for more steps...

Step 3.3.1

Simplify each term.

Tap for more steps...

Step 3.3.1.1

Multiply $p$ by $p$ by adding the exponents.

Tap for more steps...

Step 3.3.1.1.1

Move $p$ .

$\frac{p^{2} + 4 - 2 (p \cdot p) - 2 \cdot - 1 p}{{(p^{2} + 4)}^{2}}$

Step 3.3.1.1.2

Multiply $p$ by $p$ .

$\frac{p^{2} + 4 - 2 p^{2} - 2 \cdot - 1 p}{{(p^{2} + 4)}^{2}}$

Step 3.3.1.2

Multiply $- 2$ by $- 1$ .

$\frac{p^{2} + 4 - 2 p^{2} + 2 p}{{(p^{2} + 4)}^{2}}$

Step 3.3.2

Subtract $2 p^{2}$ from $p^{2}$ .

$\frac{- p^{2} + 4 + 2 p}{{(p^{2} + 4)}^{2}}$

Step 3.4

Reorder terms.

$\frac{- p^{2} + 2 p + 4}{{(p^{2} + 4)}^{2}}$

Step 3.5

Factor $- 1$ out of $- p^{2}$ .

$\frac{- (p^{2}) + 2 p + 4}{{(p^{2} + 4)}^{2}}$

Step 3.6

Factor $- 1$ out of $2 p$ .

$\frac{- (p^{2}) - (- 2 p) + 4}{{(p^{2} + 4)}^{2}}$

Step 3.7

Factor $- 1$ out of $- (p^{2}) - (- 2 p)$ .

$\frac{- (p^{2} - 2 p) + 4}{{(p^{2} + 4)}^{2}}$

Step 3.8

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{- (p^{2} - 2 p) - 1 (- 4)}{{(p^{2} + 4)}^{2}}$

Step 3.9

Factor $- 1$ out of $- (p^{2} - 2 p) - 1 (- 4)$ .

$\frac{- (p^{2} - 2 p - 4)}{{(p^{2} + 4)}^{2}}$

Step 3.10

Rewrite $- (p^{2} - 2 p - 4)$ as $- 1 (p^{2} - 2 p - 4)$ .

$\frac{- 1 (p^{2} - 2 p - 4)}{{(p^{2} + 4)}^{2}}$

Step 3.11

Move the negative in front of the fraction.

$- \frac{p^{2} - 2 p - 4}{{(p^{2} + 4)}^{2}}$