Calculus Examples

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

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4 x}{{(x^{2} + 1)}^{2}}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{x}{{(x^{2} + 1)}^{2}}]$ .

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

Step 2

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 $g (x) = {(x^{2} + 1)}^{2}$ .

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

Step 3

Differentiate using the Power Rule.

Tap for more steps...

Step 3.1

Multiply the exponents in ${({(x^{2} + 1)}^{2})}^{2}$ .

Tap for more steps...

Step 3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.1.2

Multiply $2$ by $2$ .

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

Step 3.2

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

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

Step 3.3

Multiply ${(x^{2} + 1)}^{2}$ by $1$ .

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

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} + 1$ .

Tap for more steps...

Step 4.1

To apply the Chain Rule, set $u$ as $x^{2} + 1$ .

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

Step 4.2

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

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

Step 4.3

Replace all occurrences of $u$ with $x^{2} + 1$ .

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

Step 5

Simplify with factoring out.

Tap for more steps...

Step 5.1

Multiply $2$ by $- 1$ .

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

Step 5.2

Factor $x^{2} + 1$ out of ${(x^{2} + 1)}^{2} - 2 x ((x^{2} + 1) \frac{d}{d x} [x^{2} + 1])$ .

Tap for more steps...

Step 5.2.1

Factor $x^{2} + 1$ out of ${(x^{2} + 1)}^{2}$ .

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

Step 5.2.2

Factor $x^{2} + 1$ out of $- 2 x ((x^{2} + 1) \frac{d}{d x} [x^{2} + 1])$ .

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

Step 5.2.3

Factor $x^{2} + 1$ out of $(x^{2} + 1) (x^{2} + 1) + (x^{2} + 1) (- 2 x (\frac{d}{d x} [x^{2} + 1]))$ .

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

Step 6

Cancel the common factors.

Tap for more steps...

Step 6.1

Factor $x^{2} + 1$ out of ${(x^{2} + 1)}^{4}$ .

$4 \frac{(x^{2} + 1) (x^{2} + 1 - 2 x (\frac{d}{d x} [x^{2} + 1]))}{(x^{2} + 1) {(x^{2} + 1)}^{3}}$

Step 6.2

Cancel the common factor.

$4 \frac{(x^{2} + 1) (x^{2} + 1 - 2 x (\frac{d}{d x} [x^{2} + 1]))}{(x^{2} + 1) {(x^{2} + 1)}^{3}}$

Step 6.3

Rewrite the expression.

$4 \frac{x^{2} + 1 - 2 x (\frac{d}{d x} [x^{2} + 1])}{{(x^{2} + 1)}^{3}}$

Step 7

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$4 \frac{x^{2} + 1 - 2 x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1])}{{(x^{2} + 1)}^{3}}$

Step 8

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

$4 \frac{x^{2} + 1 - 2 x (2 x + \frac{d}{d x} [1])}{{(x^{2} + 1)}^{3}}$

Step 9

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

$4 \frac{x^{2} + 1 - 2 x (2 x + 0)}{{(x^{2} + 1)}^{3}}$

Step 10

Simplify the expression.

Tap for more steps...

Step 10.1

Add $2 x$ and $0$ .

$4 \frac{x^{2} + 1 - 2 x (2 x)}{{(x^{2} + 1)}^{3}}$

Step 10.2

Multiply $2$ by $- 2$ .

$4 \frac{x^{2} + 1 - 4 x \cdot x}{{(x^{2} + 1)}^{3}}$

Step 11

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

$4 \frac{x^{2} + 1 - 4 (x^{1} x)}{{(x^{2} + 1)}^{3}}$

Step 12

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

$4 \frac{x^{2} + 1 - 4 (x^{1} x^{1})}{{(x^{2} + 1)}^{3}}$

Step 13

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

$4 \frac{x^{2} + 1 - 4 x^{1 + 1}}{{(x^{2} + 1)}^{3}}$

Step 14

Add $1$ and $1$ .

$4 \frac{x^{2} + 1 - 4 x^{2}}{{(x^{2} + 1)}^{3}}$

Step 15

Subtract $4 x^{2}$ from $x^{2}$ .

$4 \frac{- 3 x^{2} + 1}{{(x^{2} + 1)}^{3}}$

Step 16

Combine $4$ and $\frac{- 3 x^{2} + 1}{{(x^{2} + 1)}^{3}}$ .

$\frac{4 (- 3 x^{2} + 1)}{{(x^{2} + 1)}^{3}}$

Step 17

Simplify.

Tap for more steps...

Step 17.1

Apply the distributive property.

$\frac{4 (- 3 x^{2}) + 4 \cdot 1}{{(x^{2} + 1)}^{3}}$

Step 17.2

Simplify each term.

Tap for more steps...

Step 17.2.1

Multiply $- 3$ by $4$ .

$\frac{- 12 x^{2} + 4 \cdot 1}{{(x^{2} + 1)}^{3}}$

Step 17.2.2

Multiply $4$ by $1$ .

$\frac{- 12 x^{2} + 4}{{(x^{2} + 1)}^{3}}$

Step 17.3

Factor $4$ out of $- 12 x^{2} + 4$ .

Tap for more steps...

Step 17.3.1

Factor $4$ out of $- 12 x^{2}$ .

$\frac{4 (- 3 x^{2}) + 4}{{(x^{2} + 1)}^{3}}$

Step 17.3.2

Factor $4$ out of $4$ .

$\frac{4 (- 3 x^{2}) + 4 (1)}{{(x^{2} + 1)}^{3}}$

Step 17.3.3

Factor $4$ out of $4 (- 3 x^{2}) + 4 (1)$ .

$\frac{4 (- 3 x^{2} + 1)}{{(x^{2} + 1)}^{3}}$

Step 17.4

Factor $- 1$ out of $- 3 x^{2}$ .

$\frac{4 (- (3 x^{2}) + 1)}{{(x^{2} + 1)}^{3}}$

Step 17.5

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

$\frac{4 (- (3 x^{2}) - 1 (- 1))}{{(x^{2} + 1)}^{3}}$

Step 17.6

Factor $- 1$ out of $- (3 x^{2}) - 1 (- 1)$ .

$\frac{4 (- (3 x^{2} - 1))}{{(x^{2} + 1)}^{3}}$

Step 17.7

Rewrite $- (3 x^{2} - 1)$ as $- 1 (3 x^{2} - 1)$ .

$\frac{4 (- 1 (3 x^{2} - 1))}{{(x^{2} + 1)}^{3}}$

Step 17.8

Move the negative in front of the fraction.

$- \frac{4 (3 x^{2} - 1)}{{(x^{2} + 1)}^{3}}$