Calculus Examples

$F (x) = {(10 x - 9)}^{\frac{1}{2}}$ , $x = 1$

Step 1

Find the derivative.

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Step 1.1

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 $g (x) = 10 x - 9$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $10 x - 9$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [10 x - 9]$

Step 1.1.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [10 x - 9]$

Step 1.1.3

Replace all occurrences of $u$ with $10 x - 9$ .

$\frac{1}{2} {(10 x - 9)}^{\frac{1}{2} - 1} \frac{d}{d x} [10 x - 9]$

Step 1.2

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

$\frac{1}{2} {(10 x - 9)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [10 x - 9]$

Step 1.3

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

$\frac{1}{2} {(10 x - 9)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [10 x - 9]$

Step 1.4

Combine the numerators over the common denominator.

$\frac{1}{2} {(10 x - 9)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [10 x - 9]$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(10 x - 9)}^{\frac{1 - 2}{2}} \frac{d}{d x} [10 x - 9]$

Step 1.5.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(10 x - 9)}^{\frac{- 1}{2}} \frac{d}{d x} [10 x - 9]$

Step 1.6

Combine fractions.

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Step 1.6.1

Move the negative in front of the fraction.

$\frac{1}{2} {(10 x - 9)}^{- \frac{1}{2}} \frac{d}{d x} [10 x - 9]$

Step 1.6.2

Combine $\frac{1}{2}$ and ${(10 x - 9)}^{- \frac{1}{2}}$ .

$\frac{{(10 x - 9)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [10 x - 9]$

Step 1.6.3

Move ${(10 x - 9)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(10 x - 9)}^{\frac{1}{2}}} \frac{d}{d x} [10 x - 9]$

Step 1.7

By the Sum Rule, the derivative of $10 x - 9$ with respect to $x$ is $\frac{d}{d x} [10 x] + \frac{d}{d x} [- 9]$ .

$\frac{1}{2 {(10 x - 9)}^{\frac{1}{2}}} (\frac{d}{d x} [10 x] + \frac{d}{d x} [- 9])$

Step 1.8

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$\frac{1}{2 {(10 x - 9)}^{\frac{1}{2}}} (10 \frac{d}{d x} [x] + \frac{d}{d x} [- 9])$

Step 1.9

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

$\frac{1}{2 {(10 x - 9)}^{\frac{1}{2}}} (10 \cdot 1 + \frac{d}{d x} [- 9])$

Step 1.10

Multiply $10$ by $1$ .

$\frac{1}{2 {(10 x - 9)}^{\frac{1}{2}}} (10 + \frac{d}{d x} [- 9])$

Step 1.11

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$\frac{1}{2 {(10 x - 9)}^{\frac{1}{2}}} (10 + 0)$

Step 1.12

Simplify terms.

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Step 1.12.1

Add $10$ and $0$ .

$\frac{1}{2 {(10 x - 9)}^{\frac{1}{2}}} \cdot 10$

Step 1.12.2

Combine $\frac{1}{2 {(10 x - 9)}^{\frac{1}{2}}}$ and $10$ .

$\frac{10}{2 {(10 x - 9)}^{\frac{1}{2}}}$

Step 1.12.3

Factor $2$ out of $10$ .

$\frac{2 \cdot 5}{2 {(10 x - 9)}^{\frac{1}{2}}}$

Step 1.13

Cancel the common factors.

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Step 1.13.1

Factor $2$ out of $2 {(10 x - 9)}^{\frac{1}{2}}$ .

$\frac{2 \cdot 5}{2 ({(10 x - 9)}^{\frac{1}{2}})}$

Step 1.13.2

Cancel the common factor.

$\frac{2 \cdot 5}{2 {(10 x - 9)}^{\frac{1}{2}}}$

Step 1.13.3

Rewrite the expression.

$\frac{5}{{(10 x - 9)}^{\frac{1}{2}}}$

Step 2

Replace the variable $x$ with $1$ in the expression.

$F' (1) = \frac{5}{{(10 (1) - 9)}^{\frac{1}{2}}}$

Step 3

Simplify the denominator.

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Step 3.1

Multiply $10$ by $1$ .

$F' (1) = \frac{5}{{(10 - 9)}^{\frac{1}{2}}}$

Step 3.2

Subtract $9$ from $10$ .

$F' (1) = \frac{5}{1^{\frac{1}{2}}}$

Step 3.3

One to any power is one.

$F' (1) = \frac{5}{1}$

Step 4

Divide $5$ by $1$ .

$F' (1) = 5$