Calculus Examples

$\sqrt{\frac{x^{2} - 3 x}{2 x + 1}}$

Step 1

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

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

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 $g (x) = \frac{x^{2} - 3 x}{2 x + 1}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\frac{x^{2} - 3 x}{2 x + 1}$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [\frac{x^{2} - 3 x}{2 x + 1}]$

Step 2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [\frac{x^{2} - 3 x}{2 x + 1}]$

Step 2.3

Replace all occurrences of $u$ with $\frac{x^{2} - 3 x}{2 x + 1}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

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 $f (x) = x^{2} - 3 x$ and $g (x) = 2 x + 1$ .

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

Step 9

Differentiate.

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Step 9.1

By the Sum Rule, the derivative of $x^{2} - 3 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x]$ .

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

Multiply $- 3$ by $1$ .

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

Step 9.6

By the Sum Rule, the derivative of $2 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

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

Step 9.7

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 9.8

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

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

Step 9.9

Multiply $2$ by $1$ .

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

Step 9.10

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

$\frac{1}{2} {(\frac{x^{2} - 3 x}{2 x + 1})}^{- \frac{1}{2}} \frac{(2 x + 1) (2 x - 3) - (x^{2} - 3 x) (2 + 0)}{{(2 x + 1)}^{2}}$

Step 9.11

Combine fractions.

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Step 9.11.1

Add $2$ and $0$ .

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

Step 9.11.2

Multiply $2$ by $- 1$ .

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

Step 9.11.3

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

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

Step 9.11.4

Move $2$ to the left of ${(2 x + 1)}^{2}$ .

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

Step 10

Simplify.

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Step 10.1

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 10.2

Apply the product rule to $\frac{2 x + 1}{x^{2} - 3 x}$ .

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

Step 10.3

Apply the distributive property.

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

Step 10.4

Combine terms.

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Step 10.4.1

Multiply $- 3$ by $- 2$ .

$\frac{(2 x + 1) (2 x - 3) - 2 x^{2} + 6 x}{2 {(2 x + 1)}^{2}} \cdot \frac{{(2 x + 1)}^{\frac{1}{2}}}{{(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.4.2

Multiply $\frac{(2 x + 1) (2 x - 3) - 2 x^{2} + 6 x}{2 {(2 x + 1)}^{2}}$ by $\frac{{(2 x + 1)}^{\frac{1}{2}}}{{(x^{2} - 3 x)}^{\frac{1}{2}}}$ .

$\frac{((2 x + 1) (2 x - 3) - 2 x^{2} + 6 x) {(2 x + 1)}^{\frac{1}{2}}}{2 {(2 x + 1)}^{2} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.4.3

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

$\frac{(2 x + 1) (2 x - 3) - 2 x^{2} + 6 x}{2 {(2 x + 1)}^{2} {(x^{2} - 3 x)}^{\frac{1}{2}} {(2 x + 1)}^{- \frac{1}{2}}}$

Step 10.4.4

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

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Step 10.4.4.1

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

$\frac{(2 x + 1) (2 x - 3) - 2 x^{2} + 6 x}{2 ({(2 x + 1)}^{- \frac{1}{2}} {(2 x + 1)}^{2}) {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.4.4.2

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

$\frac{(2 x + 1) (2 x - 3) - 2 x^{2} + 6 x}{2 {(2 x + 1)}^{- \frac{1}{2} + 2} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.4.4.3

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

$\frac{(2 x + 1) (2 x - 3) - 2 x^{2} + 6 x}{2 {(2 x + 1)}^{- \frac{1}{2} + 2 \cdot \frac{2}{2}} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.4.4.4

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

$\frac{(2 x + 1) (2 x - 3) - 2 x^{2} + 6 x}{2 {(2 x + 1)}^{- \frac{1}{2} + \frac{2 \cdot 2}{2}} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.4.4.5

Combine the numerators over the common denominator.

$\frac{(2 x + 1) (2 x - 3) - 2 x^{2} + 6 x}{2 {(2 x + 1)}^{\frac{- 1 + 2 \cdot 2}{2}} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.4.4.6

Simplify the numerator.

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Step 10.4.4.6.1

Multiply $2$ by $2$ .

$\frac{(2 x + 1) (2 x - 3) - 2 x^{2} + 6 x}{2 {(2 x + 1)}^{\frac{- 1 + 4}{2}} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.4.4.6.2

Add $- 1$ and $4$ .

$\frac{(2 x + 1) (2 x - 3) - 2 x^{2} + 6 x}{2 {(2 x + 1)}^{\frac{3}{2}} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.5

Reorder terms.

$\frac{- 2 x^{2} + (2 x + 1) (2 x - 3) + 6 x}{2 {(2 x + 1)}^{\frac{3}{2}} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.6

Simplify the numerator.

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Step 10.6.1

Expand $(2 x + 1) (2 x - 3)$ using the FOIL Method.

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Step 10.6.1.1

Apply the distributive property.

$\frac{- 2 x^{2} + 2 x (2 x - 3) + 1 (2 x - 3) + 6 x}{2 {(2 x + 1)}^{\frac{3}{2}} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.6.1.2

Apply the distributive property.

$\frac{- 2 x^{2} + 2 x (2 x) + 2 x \cdot - 3 + 1 (2 x - 3) + 6 x}{2 {(2 x + 1)}^{\frac{3}{2}} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.6.1.3

Apply the distributive property.

$\frac{- 2 x^{2} + 2 x (2 x) + 2 x \cdot - 3 + 1 (2 x) + 1 \cdot - 3 + 6 x}{2 {(2 x + 1)}^{\frac{3}{2}} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.6.2

Simplify and combine like terms.

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Step 10.6.2.1

Simplify each term.

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Step 10.6.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{- 2 x^{2} + 2 \cdot 2 x \cdot x + 2 x \cdot - 3 + 1 (2 x) + 1 \cdot - 3 + 6 x}{2 {(2 x + 1)}^{\frac{3}{2}} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.6.2.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 10.6.2.1.2.1

Move $x$ .

$\frac{- 2 x^{2} + 2 \cdot 2 (x \cdot x) + 2 x \cdot - 3 + 1 (2 x) + 1 \cdot - 3 + 6 x}{2 {(2 x + 1)}^{\frac{3}{2}} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.6.2.1.2.2

Multiply $x$ by $x$ .

$\frac{- 2 x^{2} + 2 \cdot 2 x^{2} + 2 x \cdot - 3 + 1 (2 x) + 1 \cdot - 3 + 6 x}{2 {(2 x + 1)}^{\frac{3}{2}} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.6.2.1.3

Multiply $2$ by $2$ .

$\frac{- 2 x^{2} + 4 x^{2} + 2 x \cdot - 3 + 1 (2 x) + 1 \cdot - 3 + 6 x}{2 {(2 x + 1)}^{\frac{3}{2}} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.6.2.1.4

Multiply $- 3$ by $2$ .

$\frac{- 2 x^{2} + 4 x^{2} - 6 x + 1 (2 x) + 1 \cdot - 3 + 6 x}{2 {(2 x + 1)}^{\frac{3}{2}} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.6.2.1.5

Multiply $2 x$ by $1$ .

$\frac{- 2 x^{2} + 4 x^{2} - 6 x + 2 x + 1 \cdot - 3 + 6 x}{2 {(2 x + 1)}^{\frac{3}{2}} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.6.2.1.6

Multiply $- 3$ by $1$ .

$\frac{- 2 x^{2} + 4 x^{2} - 6 x + 2 x - 3 + 6 x}{2 {(2 x + 1)}^{\frac{3}{2}} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.6.2.2

Add $- 6 x$ and $2 x$ .

$\frac{- 2 x^{2} + 4 x^{2} - 4 x - 3 + 6 x}{2 {(2 x + 1)}^{\frac{3}{2}} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.6.3

Add $- 2 x^{2}$ and $4 x^{2}$ .

$\frac{2 x^{2} - 4 x - 3 + 6 x}{2 {(2 x + 1)}^{\frac{3}{2}} {(x^{2} - 3 x)}^{\frac{1}{2}}}$

Step 10.6.4

Add $- 4 x$ and $6 x$ .

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