Calculus Examples

$f (x) = x + \sqrt{1 - x}$

Step 1

Differentiate.

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Step 1.1

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

$\frac{d}{d x} [x] + \frac{d}{d x} [\sqrt{1 - x}]$

Step 1.2

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

$1 + \frac{d}{d x} [\sqrt{1 - x}]$

Step 2

Evaluate $\frac{d}{d x} [\sqrt{1 - x}]$ .

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - x}$ as ${(1 - x)}^{\frac{1}{2}}$ .

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

Step 2.2

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

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Step 2.2.1

To apply the Chain Rule, set $u$ as $1 - x$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $1 - x$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

Combine the numerators over the common denominator.

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

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $- 1$ by $2$ .

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

Step 2.10.2

Subtract $2$ from $1$ .

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

Step 2.11

Move the negative in front of the fraction.

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

Step 2.12

Multiply $- 1$ by $1$ .

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

Step 2.13

Subtract $1$ from $0$ .

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

Step 2.14

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

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

Step 2.15

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

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

Step 2.16

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

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

Step 2.17

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

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

Step 2.18

Move ${(1 - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.19

Move the negative in front of the fraction.

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