Calculus Examples

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

Step 1

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 $f (x) = 1 + x^{2}$ and $g (x) = {(1 - x^{2})}^{\frac{3}{2}}$ .

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

Step 2

Differentiate.

Tap for more steps...

Step 2.1

Multiply the exponents in ${({(1 - x^{2})}^{\frac{3}{2}})}^{2}$ .

Tap for more steps...

Step 2.1.1

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

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

Step 2.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.2.1

Cancel the common factor.

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

Step 2.1.2.2

Rewrite the expression.

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

Step 2.2

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

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

Step 2.3

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

$\frac{{(1 - x^{2})}^{\frac{3}{2}} (0 + \frac{d}{d x} [x^{2}]) - (1 + x^{2}) \frac{d}{d x} [{(1 - x^{2})}^{\frac{3}{2}}]}{{(1 - x^{2})}^{3}}$

Step 2.4

Add $0$ and $\frac{d}{d x} [x^{2}]$ .

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

Step 2.5

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

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

Step 2.6

Move $2$ to the left of ${(1 - x^{2})}^{\frac{3}{2}}$ .

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

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{3}{2}}$ and $g (x) = 1 - x^{2}$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{3}{2}$ .

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

Step 3.3

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

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

Step 4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

Tap for more steps...

Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $3$ .

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

Step 8

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

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

Step 9

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

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

Step 10

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

$\frac{2 {(1 - x^{2})}^{\frac{3}{2}} x - (1 + x^{2}) (\frac{3 {(1 - x^{2})}^{\frac{1}{2}}}{2} (0 + \frac{d}{d x} [- x^{2}]))}{{(1 - x^{2})}^{3}}$

Step 11

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

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

Step 12

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

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

Step 13

Multiply.

Tap for more steps...

Step 13.1

Multiply $- 1$ by $- 1$ .

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

Step 13.2

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

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

Step 14

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

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

Step 15

Simplify terms.

Tap for more steps...

Step 15.1

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

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

Step 15.2

Multiply $3$ by $2$ .

$\frac{2 {(1 - x^{2})}^{\frac{3}{2}} x + (1 + x^{2}) \frac{6 {(1 - x^{2})}^{\frac{1}{2}}}{2} x}{{(1 - x^{2})}^{3}}$

Step 15.3

Combine $x$ and $\frac{6 {(1 - x^{2})}^{\frac{1}{2}}}{2}$ .

$\frac{2 {(1 - x^{2})}^{\frac{3}{2}} x + (1 + x^{2}) \frac{x (6 {(1 - x^{2})}^{\frac{1}{2}})}{2}}{{(1 - x^{2})}^{3}}$

Step 15.4

Factor $2$ out of $x (6 {(1 - x^{2})}^{\frac{1}{2}})$ .

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

Step 16

Cancel the common factors.

Tap for more steps...

Step 16.1

Factor $2$ out of $2$ .

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

Step 16.2

Cancel the common factor.

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

Step 16.3

Rewrite the expression.

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

Step 16.4

Divide $x (3 {(1 - x^{2})}^{\frac{1}{2}})$ by $1$ .

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

Step 17

Simplify.

Tap for more steps...

Step 17.1

Apply the distributive property.

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

Step 17.2

Simplify the numerator.

Tap for more steps...

Step 17.2.1

Simplify each term.

Tap for more steps...

Step 17.2.1.1

Multiply $x$ by $1$ .

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

Step 17.2.1.2

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 17.2.1.2.1

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 17.2.1.2.1.1

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

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

Step 17.2.1.2.1.2

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

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

Step 17.2.1.2.2

Add $2$ and $1$ .

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

Step 17.2.2

Apply the distributive property.

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

Step 17.2.3

Rewrite using the commutative property of multiplication.

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

Step 17.2.4

Rewrite using the commutative property of multiplication.

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

Step 17.3

Reorder terms.

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