Calculus Examples

\sqrt{2 x - x^{2}}

$\sqrt{2 x - x^{2}}$

Step 1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{2 x - x^{2}}

$\sqrt{2 x - x^{2}}$ as

{(2 x - x^{2})}^{\frac{1}{2}}

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

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

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

Step 2

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = x^{\frac{1}{2}}$ and

$g (x) = 2 x - x^{2}$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set

$u$ as

$2 x - x^{2}$ .

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

Step 2.2

Differentiate using the Power Rule which states that

$\frac{d}{d u} [u^{n}]$ is

$n u^{n - 1}$ where

$n = \frac{1}{2}$ .

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

Step 2.3

Replace all occurrences of

$u$ with

$2 x - x^{2}$ .

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

Step 3

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Step 4

Combine

$- 1$ and

$\frac{2}{2}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Multiply

$- 1$ by

$2$ .

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

Step 6.2

Subtract

$2$ from

$1$ .

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

Step 7

Combine fractions.

Tap for more steps...

Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

Combine

$\frac{1}{2}$ and

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

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

Step 7.3

Move

${(2 x - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

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

Step 8

By the Sum Rule, the derivative of

$2 x - x^{2}$ with respect to

$x$ is

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

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

Step 9

Since

$2$ is constant with respect to

$x$ , the derivative of

$2 x$ with respect to

$x$ is

$2 \frac{d}{d x} [x]$ .

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

Step 10

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 11

Multiply

$2$ by

$1$ .

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

Step 12

Since

$- 1$ is constant with respect to

$x$ , the derivative of

$- x^{2}$ with respect to

$x$ is

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

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

Step 13

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 2$ .

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

Step 14

Multiply

$2$ by

$- 1$ .

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

Step 15

Simplify.

Tap for more steps...

Step 15.1

Reorder the factors of

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

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

Step 15.2

Multiply

$2 - 2 x$ by

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

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

Step 15.3

Factor

$2$ out of

$2$ .

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

Step 15.4

Factor

$2$ out of

$- 2 x$ .

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

Step 15.5

Factor

$2$ out of

$2 (1) + 2 (- x)$ .

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

Step 15.6

Cancel the common factors.

Tap for more steps...

Step 15.6.1

Factor

$2$ out of

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

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

Step 15.6.2

Cancel the common factor.

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

Step 15.6.3

Rewrite the expression.

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