Calculus Examples

$y = \sqrt{\frac{x - 1}{x^{4} + 1}}$

Step 1

Let $y = f (x)$ , take the natural logarithm of both sides $\ln (y) = \ln (f (x))$ .

$\ln (y) = \ln (\sqrt{\frac{x - 1}{x^{4} + 1}})$

Step 2

Expand the right hand side.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Expand $\ln ({(\frac{x - 1}{x^{4} + 1})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 2.3

Rewrite $\ln (\frac{x - 1}{x^{4} + 1})$ as $\ln (x - 1) - \ln (x^{4} + 1)$ .

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

Step 3

Differentiate the expression using the chain rule, keeping in mind that $y$ is a function of $x$ .

Tap for more steps...

Step 3.1

Differentiate the left hand side $\ln (y)$ using the chain rule.

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

Step 3.2

Differentiate the right hand side.

Tap for more steps...

Step 3.2.1

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

$\frac{y'}{y} = \frac{d}{d x} [\frac{1}{2} (\ln (x - 1) - \ln (x^{4} + 1))]$

Step 3.2.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{y'}{y} = \frac{d}{d x} [\frac{1}{2} \ln (\frac{x - 1}{x^{4} + 1})]$

Step 3.2.3

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2} \ln (\frac{x - 1}{x^{4} + 1})$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [\ln (\frac{x - 1}{x^{4} + 1})]$ .

$\frac{y'}{y} = \frac{1}{2} \frac{d}{d x} [\ln (\frac{x - 1}{x^{4} + 1})]$

Step 3.2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \frac{x - 1}{x^{4} + 1}$ .

Tap for more steps...

Step 3.2.4.1

To apply the Chain Rule, set $u$ as $\frac{x - 1}{x^{4} + 1}$ .

$\frac{y'}{y} = \frac{1}{2} (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\frac{x - 1}{x^{4} + 1}])$

Step 3.2.4.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$\frac{y'}{y} = \frac{1}{2} (\frac{1}{u} \frac{d}{d x} [\frac{x - 1}{x^{4} + 1}])$

Step 3.2.4.3

Replace all occurrences of $u$ with $\frac{x - 1}{x^{4} + 1}$ .

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

Step 3.2.5

Multiply by the reciprocal of the fraction to divide by $\frac{x - 1}{x^{4} + 1}$ .

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

Step 3.2.6

Combine fractions.

Tap for more steps...

Step 3.2.6.1

Multiply $\frac{x^{4} + 1}{x - 1}$ by $1$ .

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

Step 3.2.6.2

Multiply $\frac{x^{4} + 1}{x - 1}$ by $\frac{1}{2}$ .

$\frac{y'}{y} = \frac{x^{4} + 1}{(x - 1) \cdot 2} \frac{d}{d x} [\frac{x - 1}{x^{4} + 1}]$

Step 3.2.6.3

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

$\frac{y'}{y} = \frac{x^{4} + 1}{2 \cdot (x - 1)} \frac{d}{d x} [\frac{x - 1}{x^{4} + 1}]$

Step 3.2.7

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 $f (x) = x - 1$ and $g (x) = x^{4} + 1$ .

$\frac{y'}{y} = \frac{x^{4} + 1}{2 (x - 1)} \cdot \frac{(x^{4} + 1) \frac{d}{d x} [x - 1] - (x - 1) \frac{d}{d x} [x^{4} + 1]}{{(x^{4} + 1)}^{2}}$

Step 3.2.8

Differentiate.

Tap for more steps...

Step 3.2.8.1

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

$\frac{y'}{y} = \frac{x^{4} + 1}{2 (x - 1)} \cdot \frac{(x^{4} + 1) (\frac{d}{d x} [x] + \frac{d}{d x} [- 1]) - (x - 1) \frac{d}{d x} [x^{4} + 1]}{{(x^{4} + 1)}^{2}}$

Step 3.2.8.2

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

$\frac{y'}{y} = \frac{x^{4} + 1}{2 (x - 1)} \cdot \frac{(x^{4} + 1) (1 + \frac{d}{d x} [- 1]) - (x - 1) \frac{d}{d x} [x^{4} + 1]}{{(x^{4} + 1)}^{2}}$

Step 3.2.8.3

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

$\frac{y'}{y} = \frac{x^{4} + 1}{2 (x - 1)} \cdot \frac{(x^{4} + 1) (1 + 0) - (x - 1) \frac{d}{d x} [x^{4} + 1]}{{(x^{4} + 1)}^{2}}$

Step 3.2.8.4

Simplify the expression.

Tap for more steps...

Step 3.2.8.4.1

Add $1$ and $0$ .

$\frac{y'}{y} = \frac{x^{4} + 1}{2 (x - 1)} \cdot \frac{(x^{4} + 1) \cdot 1 - (x - 1) \frac{d}{d x} [x^{4} + 1]}{{(x^{4} + 1)}^{2}}$

Step 3.2.8.4.2

Multiply $x^{4} + 1$ by $1$ .

$\frac{y'}{y} = \frac{x^{4} + 1}{2 (x - 1)} \cdot \frac{x^{4} + 1 - (x - 1) \frac{d}{d x} [x^{4} + 1]}{{(x^{4} + 1)}^{2}}$

Step 3.2.8.5

By the Sum Rule, the derivative of $x^{4} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [1]$ .

$\frac{y'}{y} = \frac{x^{4} + 1}{2 (x - 1)} \cdot \frac{x^{4} + 1 - (x - 1) (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [1])}{{(x^{4} + 1)}^{2}}$

Step 3.2.8.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{y'}{y} = \frac{x^{4} + 1}{2 (x - 1)} \cdot \frac{x^{4} + 1 - (x - 1) (4 x^{3} + \frac{d}{d x} [1])}{{(x^{4} + 1)}^{2}}$

Step 3.2.8.7

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

$\frac{y'}{y} = \frac{x^{4} + 1}{2 (x - 1)} \cdot \frac{x^{4} + 1 - (x - 1) (4 x^{3} + 0)}{{(x^{4} + 1)}^{2}}$

Step 3.2.8.8

Combine fractions.

Tap for more steps...

Step 3.2.8.8.1

Add $4 x^{3}$ and $0$ .

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

Step 3.2.8.8.2

Multiply $4$ by $- 1$ .

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

Step 3.2.8.8.3

Multiply $\frac{x^{4} + 1}{2 (x - 1)}$ by $\frac{x^{4} + 1 - 4 (x - 1) x^{3}}{{(x^{4} + 1)}^{2}}$ .

$\frac{y'}{y} = \frac{(x^{4} + 1) (x^{4} + 1 - 4 (x - 1) x^{3})}{2 (x - 1) {(x^{4} + 1)}^{2}}$

Step 3.2.9

Cancel the common factors.

Tap for more steps...

Step 3.2.9.1

Factor $x^{4} + 1$ out of $2 (x - 1) {(x^{4} + 1)}^{2}$ .

$\frac{y'}{y} = \frac{(x^{4} + 1) (x^{4} + 1 - 4 (x - 1) x^{3})}{(x^{4} + 1) (2 (x - 1) (x^{4} + 1))}$

Step 3.2.9.2

Cancel the common factor.

$\frac{y'}{y} = \frac{(x^{4} + 1) (x^{4} + 1 - 4 (x - 1) x^{3})}{(x^{4} + 1) (2 (x - 1) (x^{4} + 1))}$

Step 3.2.9.3

Rewrite the expression.

$\frac{y'}{y} = \frac{x^{4} + 1 - 4 (x - 1) x^{3}}{2 (x - 1) (x^{4} + 1)}$

Step 3.2.10

Simplify.

Tap for more steps...

Step 3.2.10.1

Apply the distributive property.

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

Step 3.2.10.2

Apply the distributive property.

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

Step 3.2.10.3

Apply the distributive property.

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

Step 3.2.10.4

Simplify the numerator.

Tap for more steps...

Step 3.2.10.4.1

Simplify each term.

Tap for more steps...

Step 3.2.10.4.1.1

Multiply $x$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 3.2.10.4.1.1.1

Move $x^{3}$ .

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

Step 3.2.10.4.1.1.2

Multiply $x^{3}$ by $x$ .

Tap for more steps...

Step 3.2.10.4.1.1.2.1

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

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

Step 3.2.10.4.1.1.2.2

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

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

Step 3.2.10.4.1.1.3

Add $3$ and $1$ .

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

Step 3.2.10.4.1.2

Multiply $- 4$ by $- 1$ .

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

Step 3.2.10.4.2

Subtract $4 x^{4}$ from $x^{4}$ .

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

Step 3.2.10.5

Multiply $2$ by $- 1$ .

$\frac{y'}{y} = \frac{- 3 x^{4} + 1 + 4 x^{3}}{(2 x - 2) (x^{4} + 1)}$

Step 3.2.10.6

Reorder terms.

$\frac{y'}{y} = \frac{- 3 x^{4} + 4 x^{3} + 1}{(2 x - 2) (x^{4} + 1)}$

Step 3.2.10.7

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

Tap for more steps...

Step 3.2.10.7.1

Factor $2$ out of $2 x$ .

$\frac{y'}{y} = \frac{- 3 x^{4} + 4 x^{3} + 1}{(2 (x) - 2) (x^{4} + 1)}$

Step 3.2.10.7.2

Factor $2$ out of $- 2$ .

$\frac{y'}{y} = \frac{- 3 x^{4} + 4 x^{3} + 1}{(2 (x) + 2 (- 1)) (x^{4} + 1)}$

Step 3.2.10.7.3

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

$\frac{y'}{y} = \frac{- 3 x^{4} + 4 x^{3} + 1}{2 (x - 1) (x^{4} + 1)}$

Step 3.2.10.8

Factor $- 1$ out of $- 3 x^{4}$ .

$\frac{y'}{y} = \frac{- (3 x^{4}) + 4 x^{3} + 1}{2 (x - 1) (x^{4} + 1)}$

Step 3.2.10.9

Factor $- 1$ out of $4 x^{3}$ .

$\frac{y'}{y} = \frac{- (3 x^{4}) - (- 4 x^{3}) + 1}{2 (x - 1) (x^{4} + 1)}$

Step 3.2.10.10

Factor $- 1$ out of $- (3 x^{4}) - (- 4 x^{3})$ .

$\frac{y'}{y} = \frac{- (3 x^{4} - 4 x^{3}) + 1}{2 (x - 1) (x^{4} + 1)}$

Step 3.2.10.11

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

$\frac{y'}{y} = \frac{- (3 x^{4} - 4 x^{3}) - 1 (- 1)}{2 (x - 1) (x^{4} + 1)}$

Step 3.2.10.12

Factor $- 1$ out of $- (3 x^{4} - 4 x^{3}) - 1 (- 1)$ .

$\frac{y'}{y} = \frac{- (3 x^{4} - 4 x^{3} - 1)}{2 (x - 1) (x^{4} + 1)}$

Step 3.2.10.13

Rewrite $- (3 x^{4} - 4 x^{3} - 1)$ as $- 1 (3 x^{4} - 4 x^{3} - 1)$ .

$\frac{y'}{y} = \frac{- 1 (3 x^{4} - 4 x^{3} - 1)}{2 (x - 1) (x^{4} + 1)}$

Step 3.2.10.14

Move the negative in front of the fraction.

$\frac{y'}{y} = - \frac{3 x^{4} - 4 x^{3} - 1}{2 (x - 1) (x^{4} + 1)}$

Step 4

Isolate $y'$ and substitute the original function for $y$ in the right hand side.

$y' = - \frac{3 x^{4} - 4 x^{3} - 1}{2 (x - 1) (x^{4} + 1)} \sqrt{\frac{x - 1}{x^{4} + 1}}$

Step 5

Simplify the right hand side.

Tap for more steps...

Step 5.1

Rewrite $\sqrt{\frac{x - 1}{x^{4} + 1}}$ as $\frac{\sqrt{x - 1}}{\sqrt{x^{4} + 1}}$ .

$y' = - \frac{3 x^{4} - 4 x^{3} - 1}{2 (x - 1) (x^{4} + 1)} \cdot \frac{\sqrt{x - 1}}{\sqrt{x^{4} + 1}}$

Step 5.2

Multiply $\frac{\sqrt{x - 1}}{\sqrt{x^{4} + 1}}$ by $\frac{\sqrt{x^{4} + 1}}{\sqrt{x^{4} + 1}}$ .

$y' = - \frac{3 x^{4} - 4 x^{3} - 1}{2 (x - 1) (x^{4} + 1)} (\frac{\sqrt{x - 1}}{\sqrt{x^{4} + 1}} \cdot \frac{\sqrt{x^{4} + 1}}{\sqrt{x^{4} + 1}})$

Step 5.3

Combine and simplify the denominator.

Tap for more steps...

Step 5.3.1

Multiply $\frac{\sqrt{x - 1}}{\sqrt{x^{4} + 1}}$ by $\frac{\sqrt{x^{4} + 1}}{\sqrt{x^{4} + 1}}$ .

$y' = - \frac{3 x^{4} - 4 x^{3} - 1}{2 (x - 1) (x^{4} + 1)} \cdot \frac{\sqrt{x - 1} \sqrt{x^{4} + 1}}{\sqrt{x^{4} + 1} \sqrt{x^{4} + 1}}$

Step 5.3.2

Raise $\sqrt{x^{4} + 1}$ to the power of $1$ .

$y' = - \frac{3 x^{4} - 4 x^{3} - 1}{2 (x - 1) (x^{4} + 1)} \cdot \frac{\sqrt{x - 1} \sqrt{x^{4} + 1}}{{\sqrt{x^{4} + 1}}^{1} \sqrt{x^{4} + 1}}$

Step 5.3.3

Raise $\sqrt{x^{4} + 1}$ to the power of $1$ .

$y' = - \frac{3 x^{4} - 4 x^{3} - 1}{2 (x - 1) (x^{4} + 1)} \cdot \frac{\sqrt{x - 1} \sqrt{x^{4} + 1}}{{\sqrt{x^{4} + 1}}^{1} {\sqrt{x^{4} + 1}}^{1}}$

Step 5.3.4

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

$y' = - \frac{3 x^{4} - 4 x^{3} - 1}{2 (x - 1) (x^{4} + 1)} \cdot \frac{\sqrt{x - 1} \sqrt{x^{4} + 1}}{{\sqrt{x^{4} + 1}}^{1 + 1}}$

Step 5.3.5

Add $1$ and $1$ .

$y' = - \frac{3 x^{4} - 4 x^{3} - 1}{2 (x - 1) (x^{4} + 1)} \cdot \frac{\sqrt{x - 1} \sqrt{x^{4} + 1}}{{\sqrt{x^{4} + 1}}^{2}}$

Step 5.3.6

Rewrite ${\sqrt{x^{4} + 1}}^{2}$ as $x^{4} + 1$ .

Tap for more steps...

Step 5.3.6.1

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

$y' = - \frac{3 x^{4} - 4 x^{3} - 1}{2 (x - 1) (x^{4} + 1)} \cdot \frac{\sqrt{x - 1} \sqrt{x^{4} + 1}}{{({(x^{4} + 1)}^{\frac{1}{2}})}^{2}}$

Step 5.3.6.2

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

$y' = - \frac{3 x^{4} - 4 x^{3} - 1}{2 (x - 1) (x^{4} + 1)} \cdot \frac{\sqrt{x - 1} \sqrt{x^{4} + 1}}{{(x^{4} + 1)}^{\frac{1}{2} \cdot 2}}$

Step 5.3.6.3

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

$y' = - \frac{3 x^{4} - 4 x^{3} - 1}{2 (x - 1) (x^{4} + 1)} \cdot \frac{\sqrt{x - 1} \sqrt{x^{4} + 1}}{{(x^{4} + 1)}^{\frac{2}{2}}}$

Step 5.3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.3.6.4.1

Cancel the common factor.

$y' = - \frac{3 x^{4} - 4 x^{3} - 1}{2 (x - 1) (x^{4} + 1)} \cdot \frac{\sqrt{x - 1} \sqrt{x^{4} + 1}}{{(x^{4} + 1)}^{\frac{2}{2}}}$

Step 5.3.6.4.2

Rewrite the expression.

$y' = - \frac{3 x^{4} - 4 x^{3} - 1}{2 (x - 1) (x^{4} + 1)} \cdot \frac{\sqrt{x - 1} \sqrt{x^{4} + 1}}{{(x^{4} + 1)}^{1}}$

Step 5.3.6.5

Simplify.

$y' = - \frac{3 x^{4} - 4 x^{3} - 1}{2 (x - 1) (x^{4} + 1)} \cdot \frac{\sqrt{x - 1} \sqrt{x^{4} + 1}}{x^{4} + 1}$

Step 5.4

Combine using the product rule for radicals.

$y' = - \frac{3 x^{4} - 4 x^{3} - 1}{2 (x - 1) (x^{4} + 1)} \cdot \frac{\sqrt{(x - 1) (x^{4} + 1)}}{x^{4} + 1}$

Step 5.5

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

Tap for more steps...

Step 5.5.1

Multiply $\frac{\sqrt{(x - 1) (x^{4} + 1)}}{x^{4} + 1}$ by $\frac{3 x^{4} - 4 x^{3} - 1}{2 (x - 1) (x^{4} + 1)}$ .

$y' = - \frac{\sqrt{(x - 1) (x^{4} + 1)} (3 x^{4} - 4 x^{3} - 1)}{(x^{4} + 1) (2 (x - 1) (x^{4} + 1))}$

Step 5.5.2

Raise $x^{4} + 1$ to the power of $1$ .

$y' = - \frac{\sqrt{(x - 1) (x^{4} + 1)} (3 x^{4} - 4 x^{3} - 1)}{2 (x - 1) ({(x^{4} + 1)}^{1} (x^{4} + 1))}$

Step 5.5.3

Raise $x^{4} + 1$ to the power of $1$ .

$y' = - \frac{\sqrt{(x - 1) (x^{4} + 1)} (3 x^{4} - 4 x^{3} - 1)}{2 (x - 1) ({(x^{4} + 1)}^{1} {(x^{4} + 1)}^{1})}$

Step 5.5.4

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

$y' = - \frac{\sqrt{(x - 1) (x^{4} + 1)} (3 x^{4} - 4 x^{3} - 1)}{2 (x - 1) {(x^{4} + 1)}^{1 + 1}}$

Step 5.5.5

Add $1$ and $1$ .

$y' = - \frac{\sqrt{(x - 1) (x^{4} + 1)} (3 x^{4} - 4 x^{3} - 1)}{2 (x - 1) {(x^{4} + 1)}^{2}}$