Calculus Examples

$\sqrt{20 x - x^{7}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{20 x - x^{7}}$ as ${(20 x - x^{7})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(20 x - x^{7})}^{\frac{1}{2}}]$

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 $g (x) = 20 x - x^{7}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $20 x - x^{7}$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [20 x - x^{7}]$

Step 2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [20 x - x^{7}]$

Step 2.3

Replace all occurrences of $u$ with $20 x - x^{7}$ .

$\frac{1}{2} {(20 x - x^{7})}^{\frac{1}{2} - 1} \frac{d}{d x} [20 x - x^{7}]$

Step 3

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

$\frac{1}{2} {(20 x - x^{7})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [20 x - x^{7}]$

Step 4

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

$\frac{1}{2} {(20 x - x^{7})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [20 x - x^{7}]$

Step 5

Combine the numerators over the common denominator.

$\frac{1}{2} {(20 x - x^{7})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [20 x - x^{7}]$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(20 x - x^{7})}^{\frac{1 - 2}{2}} \frac{d}{d x} [20 x - x^{7}]$

Step 6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(20 x - x^{7})}^{\frac{- 1}{2}} \frac{d}{d x} [20 x - x^{7}]$

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(20 x - x^{7})}^{- \frac{1}{2}} \frac{d}{d x} [20 x - x^{7}]$

Step 7.2

Combine $\frac{1}{2}$ and ${(20 x - x^{7})}^{- \frac{1}{2}}$ .

$\frac{{(20 x - x^{7})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [20 x - x^{7}]$

Step 7.3

Move ${(20 x - x^{7})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(20 x - x^{7})}^{\frac{1}{2}}} \frac{d}{d x} [20 x - x^{7}]$

Step 8

By the Sum Rule, the derivative of $20 x - x^{7}$ with respect to $x$ is $\frac{d}{d x} [20 x] + \frac{d}{d x} [- x^{7}]$ .

$\frac{1}{2 {(20 x - x^{7})}^{\frac{1}{2}}} (\frac{d}{d x} [20 x] + \frac{d}{d x} [- x^{7}])$

Step 9

Since $20$ is constant with respect to $x$ , the derivative of $20 x$ with respect to $x$ is $20 \frac{d}{d x} [x]$ .

$\frac{1}{2 {(20 x - x^{7})}^{\frac{1}{2}}} (20 \frac{d}{d x} [x] + \frac{d}{d x} [- x^{7}])$

Step 10

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

$\frac{1}{2 {(20 x - x^{7})}^{\frac{1}{2}}} (20 \cdot 1 + \frac{d}{d x} [- x^{7}])$

Step 11

Multiply $20$ by $1$ .

$\frac{1}{2 {(20 x - x^{7})}^{\frac{1}{2}}} (20 + \frac{d}{d x} [- x^{7}])$

Step 12

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{7}$ with respect to $x$ is $- \frac{d}{d x} [x^{7}]$ .

$\frac{1}{2 {(20 x - x^{7})}^{\frac{1}{2}}} (20 - \frac{d}{d x} [x^{7}])$

Step 13

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$\frac{1}{2 {(20 x - x^{7})}^{\frac{1}{2}}} (20 - (7 x^{6}))$

Step 14

Multiply $7$ by $- 1$ .

$\frac{1}{2 {(20 x - x^{7})}^{\frac{1}{2}}} (20 - 7 x^{6})$

Step 15

Simplify.

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Step 15.1

Reorder the factors of $\frac{1}{2 {(20 x - x^{7})}^{\frac{1}{2}}} (20 - 7 x^{6})$ .

$(20 - 7 x^{6}) \frac{1}{2 {(20 x - x^{7})}^{\frac{1}{2}}}$

Step 15.2

Multiply $20 - 7 x^{6}$ by $\frac{1}{2 {(20 x - x^{7})}^{\frac{1}{2}}}$ .

$\frac{20 - 7 x^{6}}{2 {(20 x - x^{7})}^{\frac{1}{2}}}$