Calculus Examples

$y = \sqrt{\frac{4 x^{2}}{2 - 2 x}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{4 x^{2}}{2 - 2 x}}$ as ${(\frac{4 x^{2}}{2 - 2 x})}^{\frac{1}{2}}$ .

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

Step 2

Differentiate both sides of the equation.

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

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Cancel the common factor of $4$ and $2 - 2 x$ .

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Step 4.1.1

Factor $2$ out of $4 x^{2}$ .

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

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $2$ out of $2$ .

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

Step 4.1.2.2

Factor $2$ out of $- 2 x$ .

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

Step 4.1.2.3

Factor $2$ out of $2 (1) + 2 (- x)$ .

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

Step 4.1.2.4

Cancel the common factor.

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

Step 4.1.2.5

Rewrite the expression.

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

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Step 4.2.3.1

Factor $2$ out of $4 x^{2}$ .

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

Step 4.2.3.2

Cancel the common factors.

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Step 4.2.3.2.1

Factor $2$ out of $2$ .

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

Step 4.2.3.2.2

Factor $2$ out of $- 2 x$ .

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

Step 4.2.3.2.3

Factor $2$ out of $2 (1) + 2 (- x)$ .

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

Step 4.2.3.2.4

Cancel the common factor.

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

Step 4.2.3.2.5

Rewrite the expression.

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

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

Step 4.6.2

Subtract $2$ from $1$ .

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

Step 4.7

Differentiate using the Constant Multiple Rule.

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Step 4.7.1

Move the negative in front of the fraction.

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

Step 4.7.2

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2 x^{2}}{1 - x}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{x^{2}}{1 - x}]$ .

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

Step 4.7.3

Simplify terms.

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Step 4.7.3.1

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

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

Step 4.7.3.2

Cancel the common factor of $2$ .

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Step 4.7.3.2.1

Cancel the common factor.

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

Step 4.7.3.2.2

Rewrite the expression.

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

Step 4.7.3.3

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

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

Step 4.8

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

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

Step 4.9

Differentiate.

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Step 4.9.1

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

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

Step 4.9.2

Move $2$ to the left of $1 - x$ .

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

Step 4.9.3

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

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

Step 4.9.4

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

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

Step 4.9.5

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

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

Step 4.9.6

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

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

Step 4.9.7

Multiply.

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Step 4.9.7.1

Multiply $- 1$ by $- 1$ .

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

Step 4.9.7.2

Multiply $x^{2}$ by $1$ .

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

Step 4.9.8

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

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

Step 4.9.9

Multiply $x^{2}$ by $1$ .

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

Step 4.10

Simplify.

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Step 4.10.1

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

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

Step 4.10.2

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

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

Step 4.10.3

Apply the product rule to $2 x^{2}$ .

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

Step 4.10.4

Apply the distributive property.

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

Step 4.10.5

Apply the distributive property.

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

Step 4.10.6

Combine terms.

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Step 4.10.6.1

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

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Step 4.10.6.1.1

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

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

Step 4.10.6.1.2

Cancel the common factor of $2$ .

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Step 4.10.6.1.2.1

Cancel the common factor.

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

Step 4.10.6.1.2.2

Rewrite the expression.

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

Step 4.10.6.2

Simplify.

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

Step 4.10.6.3

Multiply $2$ by $1$ .

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

Step 4.10.6.4

Multiply $- 1$ by $2$ .

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

Step 4.10.6.5

Raise $x$ to the power of $1$ .

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

Step 4.10.6.6

Raise $x$ to the power of $1$ .

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

Step 4.10.6.7

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

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

Step 4.10.6.8

Add $1$ and $1$ .

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

Step 4.10.6.9

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

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

Step 4.10.6.10

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

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

Step 4.10.6.11

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

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

Step 4.10.6.12

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

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Step 4.10.6.12.1

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

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

Step 4.10.6.12.2

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

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

Step 4.10.6.12.3

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

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

Step 4.10.6.12.4

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

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

Step 4.10.6.12.5

Combine the numerators over the common denominator.

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

Step 4.10.6.12.6

Simplify the numerator.

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Step 4.10.6.12.6.1

Multiply $2$ by $2$ .

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

Step 4.10.6.12.6.2

Add $- 1$ and $4$ .

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

Step 4.10.7

Reorder terms.

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

Step 4.10.8

Factor $x$ out of $- x^{2} + 2 x$ .

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Step 4.10.8.1

Factor $x$ out of $- x^{2}$ .

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

Step 4.10.8.2

Factor $x$ out of $2 x$ .

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

Step 4.10.8.3

Factor $x$ out of $x (- x) + x \cdot 2$ .

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

Step 4.10.9

Cancel the common factor.

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

Step 4.10.10

Rewrite the expression.

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

Step 4.10.11

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

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

Step 4.10.12

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 4.10.13

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

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

Step 4.10.14

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

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

Step 4.10.15

Move the negative in front of the fraction.

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

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = - \frac{x - 2}{2^{\frac{1}{2}} {(1 - x)}^{\frac{3}{2}}}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{x - 2}{2^{\frac{1}{2}} {(1 - x)}^{\frac{3}{2}}}$