Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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 $g (x) = x^{2} + 3$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

Multiply $x^{2} + 3$ by $1$ .

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Step 1.1.2.6

Simplify the expression.

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Step 1.1.2.6.1

Add $2 x$ and $0$ .

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

Step 1.1.2.6.2

Multiply $2$ by $- 1$ .

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.1.6

Add $1$ and $1$ .

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

Step 1.1.7

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

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

Step 1.1.8

Simplify.

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Step 1.1.8.1

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

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

Step 1.1.8.2

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 1.1.8.3

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

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

Step 1.1.8.4

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

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

Step 1.1.8.5

Move the negative in front of the fraction.

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$x^{2} - 3 = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Add $3$ to both sides of the equation.

$x^{2} = 3$

Step 2.3.2

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

$x = \pm \sqrt{3}$

Step 2.3.3

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

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Step 2.3.3.1

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

$x = \sqrt{3}$

Step 2.3.3.2

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

$x = - \sqrt{3}$

Step 2.3.3.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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Step 5.1

Replace the variable $x$ with $- 2.7320508$ in the expression.

$f' (- 2.7320508) = - \frac{{(- 2.7320508)}^{2} - 3}{{({(- 2.7320508)}^{2} + 3)}^{2}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Raise $- 2.7320508$ to the power of $2$ .

$f' (- 2.7320508) = - \frac{7.46410157 - 3}{{({(- 2.7320508)}^{2} + 3)}^{2}}$

Step 5.2.1.2

Subtract $3$ from $7.46410157$ .

$f' (- 2.7320508) = - \frac{4.46410157}{{({(- 2.7320508)}^{2} + 3)}^{2}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Raise $- 2.7320508$ to the power of $2$ .

$f' (- 2.7320508) = - \frac{4.46410157}{{(7.46410157 + 3)}^{2}}$

Step 5.2.2.2

Add $7.46410157$ and $3$ .

$f' (- 2.7320508) = - \frac{4.46410157}{{10.46410157}^{2}}$

Step 5.2.2.3

Raise $10.46410157$ to the power of $2$ .

$f' (- 2.7320508) = - \frac{4.46410157}{109.49742174}$

Step 5.2.3

Simplify the expression.

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Step 5.2.3.1

Divide $4.46410157$ by $109.49742174$ .

$f' (- 2.7320508) = - 1 \cdot 0.04076901$

Step 5.2.3.2

Multiply $- 1$ by $0.04076901$ .

$f' (- 2.7320508) = - 0.04076901$

Step 5.2.4

The final answer is $- 0.04076901$ .

$- 0.04076901$

Step 5.3

At $x = - 2.7320508$ the derivative is $- 0.04076901$ . Since this is negative, the function is decreasing on $(- \infty, - \sqrt{3})$ .

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

Step 6

Substitute a value from the interval $(- 1.7320508, \sqrt{3})$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

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

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

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

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

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

Step 6.2.1.2

Subtract $3$ from $0$ .

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

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

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

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

Step 6.2.2.2

Add $0$ and $3$ .

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

Step 6.2.2.3

Raise $3$ to the power of $2$ .

$f' (0) = - \frac{- 3}{9}$

Step 6.2.3

Reduce the expression by cancelling the common factors.

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Step 6.2.3.1

Cancel the common factor of $- 3$ and $9$ .

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Step 6.2.3.1.1

Factor $3$ out of $- 3$ .

$f' (0) = - \frac{3 (- 1)}{9}$

Step 6.2.3.1.2

Cancel the common factors.

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Step 6.2.3.1.2.1

Factor $3$ out of $9$ .

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

Step 6.2.3.1.2.2

Cancel the common factor.

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

Step 6.2.3.1.2.3

Rewrite the expression.

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

Step 6.2.3.2

Move the negative in front of the fraction.

$f' (0) = \frac{1}{3}$

Step 6.2.4

The final answer is $\frac{1}{3}$ .

$\frac{1}{3}$

Step 6.3

At $x = 0$ the derivative is $\frac{1}{3}$ . Since this is positive, the function is increasing on $(- 1.7320508, \sqrt{3})$ .

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

Step 7

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

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Step 7.1

Replace the variable $x$ with $2.7320508$ in the expression.

$f' (2.7320508) = - \frac{{(2.7320508)}^{2} - 3}{{({(2.7320508)}^{2} + 3)}^{2}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the numerator.

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Step 7.2.1.1

Raise $2.7320508$ to the power of $2$ .

$f' (2.7320508) = - \frac{7.46410157 - 3}{{({2.7320508}^{2} + 3)}^{2}}$

Step 7.2.1.2

Subtract $3$ from $7.46410157$ .

$f' (2.7320508) = - \frac{4.46410157}{{({2.7320508}^{2} + 3)}^{2}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Raise $2.7320508$ to the power of $2$ .

$f' (2.7320508) = - \frac{4.46410157}{{(7.46410157 + 3)}^{2}}$

Step 7.2.2.2

Add $7.46410157$ and $3$ .

$f' (2.7320508) = - \frac{4.46410157}{{10.46410157}^{2}}$

Step 7.2.2.3

Raise $10.46410157$ to the power of $2$ .

$f' (2.7320508) = - \frac{4.46410157}{109.49742174}$

Step 7.2.3

Simplify the expression.

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Step 7.2.3.1

Divide $4.46410157$ by $109.49742174$ .

$f' (2.7320508) = - 1 \cdot 0.04076901$

Step 7.2.3.2

Multiply $- 1$ by $0.04076901$ .

$f' (2.7320508) = - 0.04076901$

Step 7.2.4

The final answer is $- 0.04076901$ .

$- 0.04076901$

Step 7.3

At $x = 2.7320508$ the derivative is $- 0.04076901$ . Since this is negative, the function is decreasing on $(\sqrt{3}, \infty)$ .

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

Step 8

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

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

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

Step 9