Calculus Examples

$y = \sqrt{\frac{1 - c x}{1 + c x}}$

Step 1

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

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

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 $g (x) = \frac{1 - c x}{1 + c x}$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Move the negative in front of the fraction.

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

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 $f (x) = 1 - c x$ and $g (x) = 1 + c x$ .

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

Step 9

Differentiate.

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

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

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

Step 9.6

Multiply $- 1$ by $1$ .

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

Step 9.7

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

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

Step 9.8

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

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

Step 9.9

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

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

Step 9.10

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

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

Step 9.11

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

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

Step 9.12

Combine fractions.

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Step 9.12.1

Multiply $c$ by $1$ .

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

Step 9.12.2

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

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

Step 9.12.3

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

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

Step 10

Simplify.

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Step 10.1

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

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

Step 10.2

Apply the product rule to $\frac{1 + c x}{1 - c x}$ .

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

Step 10.3

Apply the distributive property.

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

Step 10.4

Apply the distributive property.

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

Step 10.5

Apply the distributive property.

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

Step 10.6

Combine terms.

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Step 10.6.1

Multiply $- 1$ by $1$ .

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

Step 10.6.2

Rewrite $- 1 c$ as $- c$ .

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

Step 10.6.3

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

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

Step 10.6.4

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

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

Step 10.6.5

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

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

Step 10.6.6

Add $1$ and $1$ .

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

Step 10.6.7

Move $- 1$ to the left of $c^{2} x$ .

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

Step 10.6.8

Rewrite $- 1 c^{2}$ as $- c^{2}$ .

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

Step 10.6.9

Multiply $- 1$ by $1$ .

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

Step 10.6.10

Rewrite $- 1 c$ as $- c$ .

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

Step 10.6.11

Multiply $- 1$ by $- 1$ .

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

Step 10.6.12

Multiply $c$ by $1$ .

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

Step 10.6.13

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

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

Step 10.6.14

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

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

Step 10.6.15

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

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

Step 10.6.16

Add $1$ and $1$ .

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

Step 10.6.17

Subtract $c$ from $- c$ .

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

Step 10.6.18

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

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

Step 10.6.19

Add $- 2 c$ and $0$ .

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

Step 10.6.20

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

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Step 10.6.20.1

Factor $2$ out of $- 2 c$ .

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

Step 10.6.20.2

Cancel the common factors.

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Step 10.6.20.2.1

Factor $2$ out of $2 {(1 + c x)}^{2}$ .

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

Step 10.6.20.2.2

Cancel the common factor.

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

Step 10.6.20.2.3

Rewrite the expression.

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

Step 10.6.21

Move the negative in front of the fraction.

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

Step 10.6.22

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

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

Step 10.6.23

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

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

Step 10.6.24

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

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Step 10.6.24.1

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

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

Step 10.6.24.2

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

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

Step 10.6.24.3

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

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

Step 10.6.24.4

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

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

Step 10.6.24.5

Combine the numerators over the common denominator.

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

Step 10.6.24.6

Simplify the numerator.

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Step 10.6.24.6.1

Multiply $2$ by $2$ .

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

Step 10.6.24.6.2

Add $- 1$ and $4$ .

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