Calculus Examples

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

Step 1

Write $- \frac{2 x}{{(x^{2} + 1)}^{2}}$ as a function.

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

Step 2

Find the first derivative.

Tap for more steps...

Step 2.1

Find the first derivative.

Tap for more steps...

Step 2.1.1

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

$f' (x) = - 2 \frac{d}{d x} \frac{x}{{(x^{2} + 1)}^{2}}$

Step 2.1.2

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 $g (x) = {(x^{2} + 1)}^{2}$ .

$f' (x) = - 2 \frac{{(x^{2} + 1)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} + 1)}^{2}}{{({(x^{2} + 1)}^{2})}^{2}}$

Step 2.1.3

Differentiate using the Power Rule.

Tap for more steps...

Step 2.1.3.1

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

Tap for more steps...

Step 2.1.3.1.1

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

$f' (x) = - 2 \frac{{(x^{2} + 1)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} + 1)}^{2}}{{(x^{2} + 1)}^{2 \cdot 2}}$

Step 2.1.3.1.2

Multiply $2$ by $2$ .

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

Step 2.1.3.2

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

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

Step 2.1.3.3

Multiply ${(x^{2} + 1)}^{2}$ by $1$ .

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

Step 2.1.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} + 1$ .

Tap for more steps...

Step 2.1.4.1

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

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

Step 2.1.4.2

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

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

Step 2.1.4.3

Replace all occurrences of $u$ with $x^{2} + 1$ .

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

Step 2.1.5

Simplify with factoring out.

Tap for more steps...

Step 2.1.5.1

Multiply $2$ by $- 1$ .

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

Step 2.1.5.2

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

Tap for more steps...

Step 2.1.5.2.1

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

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

Step 2.1.5.2.2

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

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

Step 2.1.5.2.3

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

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

Step 2.1.6

Cancel the common factors.

Tap for more steps...

Step 2.1.6.1

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

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

Step 2.1.6.2

Cancel the common factor.

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

Step 2.1.6.3

Rewrite the expression.

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

Step 2.1.7

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

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

Step 2.1.8

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

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

Step 2.1.9

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

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

Step 2.1.10

Simplify the expression.

Tap for more steps...

Step 2.1.10.1

Add $2 x$ and $0$ .

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

Step 2.1.10.2

Multiply $2$ by $- 2$ .

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

Step 2.1.11

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

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

Step 2.1.12

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

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

Step 2.1.13

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

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

Step 2.1.14

Add $1$ and $1$ .

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

Step 2.1.15

Subtract $4 x^{2}$ from $x^{2}$ .

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

Step 2.1.16

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

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

Step 2.1.17

Move the negative in front of the fraction.

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

Step 2.1.18

Simplify.

Tap for more steps...

Step 2.1.18.1

Apply the distributive property.

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

Step 2.1.18.2

Simplify each term.

Tap for more steps...

Step 2.1.18.2.1

Multiply $- 3$ by $2$ .

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

Step 2.1.18.2.2

Multiply $2$ by $1$ .

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

Step 2.1.18.3

Factor $2$ out of $- 6 x^{2} + 2$ .

Tap for more steps...

Step 2.1.18.3.1

Factor $2$ out of $- 6 x^{2}$ .

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

Step 2.1.18.3.2

Factor $2$ out of $2$ .

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

Step 2.1.18.3.3

Factor $2$ out of $2 (- 3 x^{2}) + 2 (1)$ .

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

Step 2.1.18.4

Factor $- 1$ out of $- 3 x^{2}$ .

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

Step 2.1.18.5

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

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

Step 2.1.18.6

Factor $- 1$ out of $- (3 x^{2}) - 1 (- 1)$ .

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

Step 2.1.18.7

Rewrite $- (3 x^{2} - 1)$ as $- 1 (3 x^{2} - 1)$ .

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

Step 2.1.18.8

Move the negative in front of the fraction.

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

Step 2.1.18.9

Multiply $- 1$ by $- 1$ .

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

Step 2.1.18.10

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

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

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{2 (3 x^{2} - 1)}{{(x^{2} + 1)}^{3}}$ .

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

Step 3

Set the first derivative equal to $0$ then solve the equation $\frac{2 (3 x^{2} - 1)}{{(x^{2} + 1)}^{3}} = 0$ .

Tap for more steps...

Step 3.1

Set the first derivative equal to $0$ .

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

Step 3.2

Set the numerator equal to zero.

$2 (3 x^{2} - 1) = 0$

Step 3.3

Solve the equation for $x$ .

Tap for more steps...

Step 3.3.1

Divide each term in $2 (3 x^{2} - 1) = 0$ by $2$ and simplify.

Tap for more steps...

Step 3.3.1.1

Divide each term in $2 (3 x^{2} - 1) = 0$ by $2$ .

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

Step 3.3.1.2

Simplify the left side.

Tap for more steps...

Step 3.3.1.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.1.2.1.1

Cancel the common factor.

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

Step 3.3.1.2.1.2

Divide $3 x^{2} - 1$ by $1$ .

$3 x^{2} - 1 = \frac{0}{2}$

Step 3.3.1.3

Simplify the right side.

Tap for more steps...

Step 3.3.1.3.1

Divide $0$ by $2$ .

$3 x^{2} - 1 = 0$

Step 3.3.2

Add $1$ to both sides of the equation.

$3 x^{2} = 1$

Step 3.3.3

Divide each term in $3 x^{2} = 1$ by $3$ and simplify.

Tap for more steps...

Step 3.3.3.1

Divide each term in $3 x^{2} = 1$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{1}{3}$

Step 3.3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.3.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{1}{3}$

Step 3.3.3.2.1.2

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

$x^{2} = \frac{1}{3}$

Step 3.3.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1}{3}}$

Step 3.3.5

Simplify $\pm \sqrt{\frac{1}{3}}$ .

Tap for more steps...

Step 3.3.5.1

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{3}}$

Step 3.3.5.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{3}}$

Step 3.3.5.3

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 3.3.5.4

Combine and simplify the denominator.

Tap for more steps...

Step 3.3.5.4.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.3.5.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 3.3.5.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 3.3.5.4.4

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

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 3.3.5.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 3.3.5.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 3.3.5.4.6.1

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

$x = \pm \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 3.3.5.4.6.2

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

$x = \pm \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 3.3.5.4.6.3

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

$x = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.3.5.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.5.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.3.5.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{3}}{3^{1}}$

Step 3.3.5.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{3}}{3}$

Step 3.3.6

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 3.3.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{3}}{3}$

Step 3.3.6.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 3.3.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3}$

Step 4

The values which make the derivative equal to $0$ are $\frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3}$ .

$\frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3}$

Step 5

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, - \frac{\sqrt{3}}{3}) \cup (- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}) \cup (\frac{\sqrt{3}}{3}, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - \frac{\sqrt{3}}{3})$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $- 1.57735026$ in the expression.

$f' (- 1.57735026) = \frac{2 (3 {(- 1.57735026)}^{2} - 1)}{{({(- 1.57735026)}^{2} + 1)}^{3}}$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Simplify the numerator.

Tap for more steps...

Step 6.2.1.1

Raise $- 1.57735026$ to the power of $2$ .

$f' (- 1.57735026) = \frac{2 (3 \cdot 2.48803384 - 1)}{{({(- 1.57735026)}^{2} + 1)}^{3}}$

Step 6.2.1.2

Multiply $3$ by $2.48803384$ .

$f' (- 1.57735026) = \frac{2 (7.46410152 - 1)}{{({(- 1.57735026)}^{2} + 1)}^{3}}$

Step 6.2.1.3

Subtract $1$ from $7.46410152$ .

$f' (- 1.57735026) = \frac{2 \cdot 6.46410152}{{({(- 1.57735026)}^{2} + 1)}^{3}}$

Step 6.2.2

Simplify the denominator.

Tap for more steps...

Step 6.2.2.1

Raise $- 1.57735026$ to the power of $2$ .

$f' (- 1.57735026) = \frac{2 \cdot 6.46410152}{{(2.48803384 + 1)}^{3}}$

Step 6.2.2.2

Add $2.48803384$ and $1$ .

$f' (- 1.57735026) = \frac{2 \cdot 6.46410152}{{3.48803384}^{3}}$

Step 6.2.2.3

Raise $3.48803384$ to the power of $3$ .

$f' (- 1.57735026) = \frac{2 \cdot 6.46410152}{42.43674549}$

Step 6.2.3

Simplify the expression.

Tap for more steps...

Step 6.2.3.1

Multiply $2$ by $6.46410152$ .

$f' (- 1.57735026) = \frac{12.92820305}{42.43674549}$

Step 6.2.3.2

Divide $12.92820305$ by $42.43674549$ .

$f' (- 1.57735026) = 0.30464643$

Step 6.2.4

The final answer is $0.30464643$ .

$0.30464643$

Step 6.3

At $x = - 1.57735026$ the derivative is $0.30464643$ . Since this is positive, the function is increasing on $(- \infty, - \frac{\sqrt{3}}{3})$ .

Increasing on $(- \infty, - \frac{\sqrt{3}}{3})$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(- 0.57735026, \frac{\sqrt{3}}{3})$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 7.1

Replace the variable $x$ with $0$ in the expression.

$f' (0) = \frac{2 (3 {(0)}^{2} - 1)}{{({(0)}^{2} + 1)}^{3}}$

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Simplify the numerator.

Tap for more steps...

Step 7.2.1.1

Raising $0$ to any positive power yields $0$ .

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

Step 7.2.1.2

Multiply $3$ by $0$ .

$f' (0) = \frac{2 (0 - 1)}{{(0^{2} + 1)}^{3}}$

Step 7.2.1.3

Subtract $1$ from $0$ .

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

Step 7.2.2

Simplify the denominator.

Tap for more steps...

Step 7.2.2.1

Raising $0$ to any positive power yields $0$ .

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

Step 7.2.2.2

Add $0$ and $1$ .

$f' (0) = \frac{2 \cdot - 1}{1^{3}}$

Step 7.2.2.3

One to any power is one.

$f' (0) = \frac{2 \cdot - 1}{1}$

Step 7.2.3

Simplify the expression.

Tap for more steps...

Step 7.2.3.1

Multiply $2$ by $- 1$ .

$f' (0) = \frac{- 2}{1}$

Step 7.2.3.2

Divide $- 2$ by $1$ .

$f' (0) = - 2$

Step 7.2.4

The final answer is $- 2$ .

$- 2$

Step 7.3

At $x = 0$ the derivative is $- 2$ . Since this is negative, the function is decreasing on $(- 0.57735026, \frac{\sqrt{3}}{3})$ .

Decreasing on $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(\frac{\sqrt{3}}{3}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 8.1

Replace the variable $x$ with $1.57735026$ in the expression.

$f' (1.57735026) = \frac{2 (3 {(1.57735026)}^{2} - 1)}{{({(1.57735026)}^{2} + 1)}^{3}}$

Step 8.2

Simplify the result.

Tap for more steps...

Step 8.2.1

Simplify the numerator.

Tap for more steps...

Step 8.2.1.1

Raise $1.57735026$ to the power of $2$ .

$f' (1.57735026) = \frac{2 (3 \cdot 2.48803384 - 1)}{{({1.57735026}^{2} + 1)}^{3}}$

Step 8.2.1.2

Multiply $3$ by $2.48803384$ .

$f' (1.57735026) = \frac{2 (7.46410152 - 1)}{{({1.57735026}^{2} + 1)}^{3}}$

Step 8.2.1.3

Subtract $1$ from $7.46410152$ .

$f' (1.57735026) = \frac{2 \cdot 6.46410152}{{({1.57735026}^{2} + 1)}^{3}}$

Step 8.2.2

Simplify the denominator.

Tap for more steps...

Step 8.2.2.1

Raise $1.57735026$ to the power of $2$ .

$f' (1.57735026) = \frac{2 \cdot 6.46410152}{{(2.48803384 + 1)}^{3}}$

Step 8.2.2.2

Add $2.48803384$ and $1$ .

$f' (1.57735026) = \frac{2 \cdot 6.46410152}{{3.48803384}^{3}}$

Step 8.2.2.3

Raise $3.48803384$ to the power of $3$ .

$f' (1.57735026) = \frac{2 \cdot 6.46410152}{42.43674549}$

Step 8.2.3

Simplify the expression.

Tap for more steps...

Step 8.2.3.1

Multiply $2$ by $6.46410152$ .

$f' (1.57735026) = \frac{12.92820305}{42.43674549}$

Step 8.2.3.2

Divide $12.92820305$ by $42.43674549$ .

$f' (1.57735026) = 0.30464643$

Step 8.2.4

The final answer is $0.30464643$ .

$0.30464643$

Step 8.3

At $x = 1.57735026$ the derivative is $0.30464643$ . Since this is positive, the function is increasing on $(\frac{\sqrt{3}}{3}, \infty)$ .

Increasing on $(\frac{\sqrt{3}}{3}, \infty)$ since $f' (x) > 0$

Step 9

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- \infty, - \frac{\sqrt{3}}{3}), (\frac{\sqrt{3}}{3}, \infty)$

Decreasing on: $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$

Step 10