Calculus Examples

${(10 x - 4)}^{\frac{1}{2}}$ , $x = 2$

Step 1

Write ${(10 x - 4)}^{\frac{1}{2}}$ as an equation.

$y = {(10 x - 4)}^{\frac{1}{2}}$

Step 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 $g (x) = 10 x - 4$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $10 x - 4$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [10 x - 4]$

Step 2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [10 x - 4]$

Step 2.3

Replace all occurrences of $u$ with $10 x - 4$ .

$\frac{1}{2} {(10 x - 4)}^{\frac{1}{2} - 1} \frac{d}{d x} [10 x - 4]$

Step 3

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

$\frac{1}{2} {(10 x - 4)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [10 x - 4]$

Step 4

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

$\frac{1}{2} {(10 x - 4)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [10 x - 4]$

Step 5

Combine the numerators over the common denominator.

$\frac{1}{2} {(10 x - 4)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [10 x - 4]$

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(10 x - 4)}^{\frac{1 - 2}{2}} \frac{d}{d x} [10 x - 4]$

Step 6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(10 x - 4)}^{\frac{- 1}{2}} \frac{d}{d x} [10 x - 4]$

Step 7

Combine fractions.

Tap for more steps...

Step 7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(10 x - 4)}^{- \frac{1}{2}} \frac{d}{d x} [10 x - 4]$

Step 7.2

Combine $\frac{1}{2}$ and ${(10 x - 4)}^{- \frac{1}{2}}$ .

$\frac{{(10 x - 4)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [10 x - 4]$

Step 7.3

Move ${(10 x - 4)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(10 x - 4)}^{\frac{1}{2}}} \frac{d}{d x} [10 x - 4]$

Step 8

By the Sum Rule, the derivative of $10 x - 4$ with respect to $x$ is $\frac{d}{d x} [10 x] + \frac{d}{d x} [- 4]$ .

$\frac{1}{2 {(10 x - 4)}^{\frac{1}{2}}} (\frac{d}{d x} [10 x] + \frac{d}{d x} [- 4])$

Step 9

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

$\frac{1}{2 {(10 x - 4)}^{\frac{1}{2}}} (10 \frac{d}{d x} [x] + \frac{d}{d x} [- 4])$

Step 10

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

$\frac{1}{2 {(10 x - 4)}^{\frac{1}{2}}} (10 \cdot 1 + \frac{d}{d x} [- 4])$

Step 11

Multiply $10$ by $1$ .

$\frac{1}{2 {(10 x - 4)}^{\frac{1}{2}}} (10 + \frac{d}{d x} [- 4])$

Step 12

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$\frac{1}{2 {(10 x - 4)}^{\frac{1}{2}}} (10 + 0)$

Step 13

Simplify terms.

Tap for more steps...

Step 13.1

Add $10$ and $0$ .

$\frac{1}{2 {(10 x - 4)}^{\frac{1}{2}}} \cdot 10$

Step 13.2

Combine $\frac{1}{2 {(10 x - 4)}^{\frac{1}{2}}}$ and $10$ .

$\frac{10}{2 {(10 x - 4)}^{\frac{1}{2}}}$

Step 13.3

Factor $2$ out of $10$ .

$\frac{2 \cdot 5}{2 {(10 x - 4)}^{\frac{1}{2}}}$

Step 14

Cancel the common factors.

Tap for more steps...

Step 14.1

Factor $2$ out of $2 {(10 x - 4)}^{\frac{1}{2}}$ .

$\frac{2 \cdot 5}{2 ({(10 x - 4)}^{\frac{1}{2}})}$

Step 14.2

Cancel the common factor.

$\frac{2 \cdot 5}{2 {(10 x - 4)}^{\frac{1}{2}}}$

Step 14.3

Rewrite the expression.

$\frac{5}{{(10 x - 4)}^{\frac{1}{2}}}$

Step 15

Evaluate the derivative at $x = 2$ .

$\frac{5}{{(10 (2) - 4)}^{\frac{1}{2}}}$

Step 16

Simplify the denominator.

Tap for more steps...

Step 16.1

Multiply $10$ by $2$ .

$\frac{5}{{(20 - 4)}^{\frac{1}{2}}}$

Step 16.2

Subtract $4$ from $20$ .

$\frac{5}{16^{\frac{1}{2}}}$

Step 16.3

Rewrite $16$ as $4^{2}$ .

$\frac{5}{{(4^{2})}^{\frac{1}{2}}}$

Step 16.4

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

$\frac{5}{4^{2 (\frac{1}{2})}}$

Step 16.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 16.5.1

Cancel the common factor.

$\frac{5}{4^{2 (\frac{1}{2})}}$

Step 16.5.2

Rewrite the expression.

$\frac{5}{4^{1}}$

Step 16.6

Evaluate the exponent.

$\frac{5}{4}$