Calculus Examples

$\frac{x}{\sqrt{7 - 3 x}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 - 3 x}$ as ${(7 - 3 x)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\frac{x}{{(7 - 3 x)}^{\frac{1}{2}}}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = {(7 - 3 x)}^{\frac{1}{2}}$ .

$\frac{{(7 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x] - x \frac{d}{d x} [{(7 - 3 x)}^{\frac{1}{2}}]}{{({(7 - 3 x)}^{\frac{1}{2}})}^{2}}$

Step 3

Multiply the exponents in ${({(7 - 3 x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.1

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

$\frac{{(7 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x] - x \frac{d}{d x} [{(7 - 3 x)}^{\frac{1}{2}}]}{{(7 - 3 x)}^{\frac{1}{2} \cdot 2}}$

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

$\frac{{(7 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x] - x \frac{d}{d x} [{(7 - 3 x)}^{\frac{1}{2}}]}{{(7 - 3 x)}^{\frac{1}{2} \cdot 2}}$

Step 3.2.2

Rewrite the expression.

$\frac{{(7 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x] - x \frac{d}{d x} [{(7 - 3 x)}^{\frac{1}{2}}]}{{(7 - 3 x)}^{1}}$

Step 4

Simplify.

$\frac{{(7 - 3 x)}^{\frac{1}{2}} \frac{d}{d x} [x] - x \frac{d}{d x} [{(7 - 3 x)}^{\frac{1}{2}}]}{7 - 3 x}$

Step 5

Differentiate using the Power Rule.

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Step 5.1

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

$\frac{{(7 - 3 x)}^{\frac{1}{2}} \cdot 1 - x \frac{d}{d x} [{(7 - 3 x)}^{\frac{1}{2}}]}{7 - 3 x}$

Step 5.2

Multiply ${(7 - 3 x)}^{\frac{1}{2}}$ by $1$ .

$\frac{{(7 - 3 x)}^{\frac{1}{2}} - x \frac{d}{d x} [{(7 - 3 x)}^{\frac{1}{2}}]}{7 - 3 x}$

Step 6

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 $g (x) = 7 - 3 x$ .

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Step 6.1

To apply the Chain Rule, set $u$ as $7 - 3 x$ .

$\frac{{(7 - 3 x)}^{\frac{1}{2}} - x (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [7 - 3 x])}{7 - 3 x}$

Step 6.2

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

$\frac{{(7 - 3 x)}^{\frac{1}{2}} - x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [7 - 3 x])}{7 - 3 x}$

Step 6.3

Replace all occurrences of $u$ with $7 - 3 x$ .

$\frac{{(7 - 3 x)}^{\frac{1}{2}} - x (\frac{1}{2} {(7 - 3 x)}^{\frac{1}{2} - 1} \frac{d}{d x} [7 - 3 x])}{7 - 3 x}$

Step 7

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

$\frac{{(7 - 3 x)}^{\frac{1}{2}} - x (\frac{1}{2} {(7 - 3 x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [7 - 3 x])}{7 - 3 x}$

Step 8

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

$\frac{{(7 - 3 x)}^{\frac{1}{2}} - x (\frac{1}{2} {(7 - 3 x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [7 - 3 x])}{7 - 3 x}$

Step 9

Combine the numerators over the common denominator.

$\frac{{(7 - 3 x)}^{\frac{1}{2}} - x (\frac{1}{2} {(7 - 3 x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [7 - 3 x])}{7 - 3 x}$

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

$\frac{{(7 - 3 x)}^{\frac{1}{2}} - x (\frac{1}{2} {(7 - 3 x)}^{\frac{1 - 2}{2}} \frac{d}{d x} [7 - 3 x])}{7 - 3 x}$

Step 10.2

Subtract $2$ from $1$ .

$\frac{{(7 - 3 x)}^{\frac{1}{2}} - x (\frac{1}{2} {(7 - 3 x)}^{\frac{- 1}{2}} \frac{d}{d x} [7 - 3 x])}{7 - 3 x}$

Step 11

Combine fractions.

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Step 11.1

Move the negative in front of the fraction.

$\frac{{(7 - 3 x)}^{\frac{1}{2}} - x (\frac{1}{2} {(7 - 3 x)}^{- \frac{1}{2}} \frac{d}{d x} [7 - 3 x])}{7 - 3 x}$

Step 11.2

Combine $\frac{1}{2}$ and ${(7 - 3 x)}^{- \frac{1}{2}}$ .

$\frac{{(7 - 3 x)}^{\frac{1}{2}} - x (\frac{{(7 - 3 x)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [7 - 3 x])}{7 - 3 x}$

Step 11.3

Move ${(7 - 3 x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(7 - 3 x)}^{\frac{1}{2}} - x (\frac{1}{2 {(7 - 3 x)}^{\frac{1}{2}}} \frac{d}{d x} [7 - 3 x])}{7 - 3 x}$

Step 11.4

Combine $\frac{1}{2 {(7 - 3 x)}^{\frac{1}{2}}}$ and $x$ .

$\frac{{(7 - 3 x)}^{\frac{1}{2}} - \frac{x}{2 {(7 - 3 x)}^{\frac{1}{2}}} \frac{d}{d x} [7 - 3 x]}{7 - 3 x}$

Step 12

By the Sum Rule, the derivative of $7 - 3 x$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [- 3 x]$ .

$\frac{{(7 - 3 x)}^{\frac{1}{2}} - \frac{x}{2 {(7 - 3 x)}^{\frac{1}{2}}} (\frac{d}{d x} [7] + \frac{d}{d x} [- 3 x])}{7 - 3 x}$

Step 13

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$\frac{{(7 - 3 x)}^{\frac{1}{2}} - \frac{x}{2 {(7 - 3 x)}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- 3 x])}{7 - 3 x}$

Step 14

Add $0$ and $\frac{d}{d x} [- 3 x]$ .

$\frac{{(7 - 3 x)}^{\frac{1}{2}} - \frac{x}{2 {(7 - 3 x)}^{\frac{1}{2}}} \frac{d}{d x} [- 3 x]}{7 - 3 x}$

Step 15

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$\frac{{(7 - 3 x)}^{\frac{1}{2}} - \frac{x}{2 {(7 - 3 x)}^{\frac{1}{2}}} (- 3 \frac{d}{d x} [x])}{7 - 3 x}$

Step 16

Combine fractions.

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Step 16.1

Multiply $- 3$ by $- 1$ .

$\frac{{(7 - 3 x)}^{\frac{1}{2}} + 3 \frac{x}{2 {(7 - 3 x)}^{\frac{1}{2}}} \frac{d}{d x} [x]}{7 - 3 x}$

Step 16.2

Combine $3$ and $\frac{x}{2 {(7 - 3 x)}^{\frac{1}{2}}}$ .

$\frac{{(7 - 3 x)}^{\frac{1}{2}} + \frac{3 x}{2 {(7 - 3 x)}^{\frac{1}{2}}} \frac{d}{d x} [x]}{7 - 3 x}$

Step 17

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

$\frac{{(7 - 3 x)}^{\frac{1}{2}} + \frac{3 x}{2 {(7 - 3 x)}^{\frac{1}{2}}} \cdot 1}{7 - 3 x}$

Step 18

Multiply $\frac{3 x}{2 {(7 - 3 x)}^{\frac{1}{2}}}$ by $1$ .

$\frac{{(7 - 3 x)}^{\frac{1}{2}} + \frac{3 x}{2 {(7 - 3 x)}^{\frac{1}{2}}}}{7 - 3 x}$

Step 19

To write ${(7 - 3 x)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(7 - 3 x)}^{\frac{1}{2}}}{2 {(7 - 3 x)}^{\frac{1}{2}}}$ .

$\frac{{(7 - 3 x)}^{\frac{1}{2}} \cdot \frac{2 {(7 - 3 x)}^{\frac{1}{2}}}{2 {(7 - 3 x)}^{\frac{1}{2}}} + \frac{3 x}{2 {(7 - 3 x)}^{\frac{1}{2}}}}{7 - 3 x}$

Step 20

Combine ${(7 - 3 x)}^{\frac{1}{2}}$ and $\frac{2 {(7 - 3 x)}^{\frac{1}{2}}}{2 {(7 - 3 x)}^{\frac{1}{2}}}$ .

$\frac{\frac{{(7 - 3 x)}^{\frac{1}{2}} (2 {(7 - 3 x)}^{\frac{1}{2}})}{2 {(7 - 3 x)}^{\frac{1}{2}}} + \frac{3 x}{2 {(7 - 3 x)}^{\frac{1}{2}}}}{7 - 3 x}$

Step 21

Combine the numerators over the common denominator.

$\frac{\frac{{(7 - 3 x)}^{\frac{1}{2}} (2 {(7 - 3 x)}^{\frac{1}{2}}) + 3 x}{2 {(7 - 3 x)}^{\frac{1}{2}}}}{7 - 3 x}$

Step 22

Multiply ${(7 - 3 x)}^{\frac{1}{2}}$ by ${(7 - 3 x)}^{\frac{1}{2}}$ by adding the exponents.

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Step 22.1

Move ${(7 - 3 x)}^{\frac{1}{2}}$ .

$\frac{\frac{{(7 - 3 x)}^{\frac{1}{2}} {(7 - 3 x)}^{\frac{1}{2}} \cdot 2 + 3 x}{2 {(7 - 3 x)}^{\frac{1}{2}}}}{7 - 3 x}$

Step 22.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{{(7 - 3 x)}^{\frac{1}{2} + \frac{1}{2}} \cdot 2 + 3 x}{2 {(7 - 3 x)}^{\frac{1}{2}}}}{7 - 3 x}$

Step 22.3

Combine the numerators over the common denominator.

$\frac{\frac{{(7 - 3 x)}^{\frac{1 + 1}{2}} \cdot 2 + 3 x}{2 {(7 - 3 x)}^{\frac{1}{2}}}}{7 - 3 x}$

Step 22.4

Add $1$ and $1$ .

$\frac{\frac{{(7 - 3 x)}^{\frac{2}{2}} \cdot 2 + 3 x}{2 {(7 - 3 x)}^{\frac{1}{2}}}}{7 - 3 x}$

Step 22.5

Divide $2$ by $2$ .

$\frac{\frac{{(7 - 3 x)}^{1} \cdot 2 + 3 x}{2 {(7 - 3 x)}^{\frac{1}{2}}}}{7 - 3 x}$

Step 23

Simplify ${(7 - 3 x)}^{1} \cdot 2$ .

$\frac{\frac{(7 - 3 x) \cdot 2 + 3 x}{2 {(7 - 3 x)}^{\frac{1}{2}}}}{7 - 3 x}$

Step 24

Move $2$ to the left of $7 - 3 x$ .

$\frac{\frac{2 \cdot (7 - 3 x) + 3 x}{2 {(7 - 3 x)}^{\frac{1}{2}}}}{7 - 3 x}$

Step 25

Rewrite $\frac{\frac{2 (7 - 3 x) + 3 x}{2 {(7 - 3 x)}^{\frac{1}{2}}}}{7 - 3 x}$ as a product.

$\frac{2 (7 - 3 x) + 3 x}{2 {(7 - 3 x)}^{\frac{1}{2}}} \cdot \frac{1}{7 - 3 x}$

Step 26

Multiply $\frac{2 (7 - 3 x) + 3 x}{2 {(7 - 3 x)}^{\frac{1}{2}}}$ by $\frac{1}{7 - 3 x}$ .

$\frac{2 (7 - 3 x) + 3 x}{2 {(7 - 3 x)}^{\frac{1}{2}} (7 - 3 x)}$

Step 27

Raise $7 - 3 x$ to the power of $1$ .

$\frac{2 (7 - 3 x) + 3 x}{2 ({(7 - 3 x)}^{1} {(7 - 3 x)}^{\frac{1}{2}})}$

Step 28

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 (7 - 3 x) + 3 x}{2 {(7 - 3 x)}^{1 + \frac{1}{2}}}$

Step 29

Write $1$ as a fraction with a common denominator.

$\frac{2 (7 - 3 x) + 3 x}{2 {(7 - 3 x)}^{\frac{2}{2} + \frac{1}{2}}}$

Step 30

Combine the numerators over the common denominator.

$\frac{2 (7 - 3 x) + 3 x}{2 {(7 - 3 x)}^{\frac{2 + 1}{2}}}$

Step 31

Add $2$ and $1$ .

$\frac{2 (7 - 3 x) + 3 x}{2 {(7 - 3 x)}^{\frac{3}{2}}}$

Step 32

Simplify.

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Step 32.1

Apply the distributive property.

$\frac{2 \cdot 7 + 2 (- 3 x) + 3 x}{2 {(7 - 3 x)}^{\frac{3}{2}}}$

Step 32.2

Simplify the numerator.

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Step 32.2.1

Simplify each term.

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Step 32.2.1.1

Multiply $2$ by $7$ .

$\frac{14 + 2 (- 3 x) + 3 x}{2 {(7 - 3 x)}^{\frac{3}{2}}}$

Step 32.2.1.2

Multiply $- 3$ by $2$ .

$\frac{14 - 6 x + 3 x}{2 {(7 - 3 x)}^{\frac{3}{2}}}$

Step 32.2.2

Add $- 6 x$ and $3 x$ .

$\frac{14 - 3 x}{2 {(7 - 3 x)}^{\frac{3}{2}}}$

Step 32.3

Reorder terms.

$\frac{- 3 x + 14}{2 {(7 - 3 x)}^{\frac{3}{2}}}$

Step 32.4

Factor $- 1$ out of $- 3 x$ .

$\frac{- (3 x) + 14}{2 {(7 - 3 x)}^{\frac{3}{2}}}$

Step 32.5

Rewrite $14$ as $- 1 (- 14)$ .

$\frac{- (3 x) - 1 (- 14)}{2 {(7 - 3 x)}^{\frac{3}{2}}}$

Step 32.6

Factor $- 1$ out of $- (3 x) - 1 (- 14)$ .

$\frac{- (3 x - 14)}{2 {(7 - 3 x)}^{\frac{3}{2}}}$

Step 32.7

Rewrite $- (3 x - 14)$ as $- 1 (3 x - 14)$ .

$\frac{- 1 (3 x - 14)}{2 {(7 - 3 x)}^{\frac{3}{2}}}$

Step 32.8

Move the negative in front of the fraction.

$- \frac{3 x - 14}{2 {(7 - 3 x)}^{\frac{3}{2}}}$