Calculus Examples

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

Step 1

Move the negative in front of the fraction.

$\frac{d}{d x} [- \frac{x}{x^{2} + 4}]$

Step 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 $f (x) = - 1$ and $g (x) = \frac{x}{x^{2} + 4}$ .

$- \frac{d}{d x} [\frac{x}{x^{2} + 4}] + \frac{x}{x^{2} + 4} \frac{d}{d x} [- 1]$

Step 3

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

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

Step 4

Differentiate.

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

Simplify the expression.

Tap for more steps...

Step 4.6.1

Add $2 x$ and $0$ .

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

Step 4.6.2

Multiply $2$ by $- 1$ .

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

Step 5

Raise $x$ to the power of $1$ .

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

Step 6

Raise $x$ to the power of $1$ .

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

Step 7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 8

Simplify by adding terms.

Tap for more steps...

Step 8.1

Add $1$ and $1$ .

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

Step 8.2

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

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

Step 9

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

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

Step 10

Simplify the expression.

Tap for more steps...

Step 10.1

Multiply $\frac{x}{x^{2} + 4}$ by $0$ .

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

Step 10.2

Add $- \frac{- x^{2} + 4}{{(x^{2} + 4)}^{2}}$ and $0$ .

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