Calculus Examples

$f (x) = \frac{x + 10}{x^{2}}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

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 $f (x) = x + 10$ and $g (x) = x^{2}$ .

$\frac{x^{2} \frac{d}{d x} [x + 10] - (x + 10) \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

Step 1.1.2

Differentiate.

Tap for more steps...

Step 1.1.2.1

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

Tap for more steps...

Step 1.1.2.1.1

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

$\frac{x^{2} \frac{d}{d x} [x + 10] - (x + 10) \frac{d}{d x} [x^{2}]}{x^{2 \cdot 2}}$

Step 1.1.2.1.2

Multiply $2$ by $2$ .

$\frac{x^{2} \frac{d}{d x} [x + 10] - (x + 10) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 1.1.2.2

By the Sum Rule, the derivative of $x + 10$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [10]$ .

$\frac{x^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [10]) - (x + 10) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 1.1.2.3

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

$\frac{x^{2} (1 + \frac{d}{d x} [10]) - (x + 10) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 1.1.2.4

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$\frac{x^{2} (1 + 0) - (x + 10) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 1.1.2.5

Simplify the expression.

Tap for more steps...

Step 1.1.2.5.1

Add $1$ and $0$ .

$\frac{x^{2} \cdot 1 - (x + 10) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 1.1.2.5.2

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

$\frac{x^{2} - (x + 10) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 1.1.2.6

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

$\frac{x^{2} - (x + 10) (2 x)}{x^{4}}$

Step 1.1.2.7

Simplify with factoring out.

Tap for more steps...

Step 1.1.2.7.1

Multiply $2$ by $- 1$ .

$\frac{x^{2} - 2 (x + 10) x}{x^{4}}$

Step 1.1.2.7.2

Factor $x$ out of $x^{2} - 2 (x + 10) x$ .

Tap for more steps...

Step 1.1.2.7.2.1

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

$\frac{x \cdot x - 2 (x + 10) x}{x^{4}}$

Step 1.1.2.7.2.2

Factor $x$ out of $- 2 (x + 10) x$ .

$\frac{x \cdot x + x (- 2 (x + 10))}{x^{4}}$

Step 1.1.2.7.2.3

Factor $x$ out of $x \cdot x + x (- 2 (x + 10))$ .

$\frac{x (x - 2 (x + 10))}{x^{4}}$

Step 1.1.3

Cancel the common factors.

Tap for more steps...

Step 1.1.3.1

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

$\frac{x (x - 2 (x + 10))}{x \cdot x^{3}}$

Step 1.1.3.2

Cancel the common factor.

$\frac{x (x - 2 (x + 10))}{x \cdot x^{3}}$

Step 1.1.3.3

Rewrite the expression.

$\frac{x - 2 (x + 10)}{x^{3}}$

Step 1.1.4

Simplify.

Tap for more steps...

Step 1.1.4.1

Apply the distributive property.

$\frac{x - 2 x - 2 \cdot 10}{x^{3}}$

Step 1.1.4.2

Simplify the numerator.

Tap for more steps...

Step 1.1.4.2.1

Multiply $- 2$ by $10$ .

$\frac{x - 2 x - 20}{x^{3}}$

Step 1.1.4.2.2

Subtract $2 x$ from $x$ .

$\frac{- x - 20}{x^{3}}$

Step 1.1.4.3

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

$\frac{- (x) - 20}{x^{3}}$

Step 1.1.4.4

Rewrite $- 20$ as $- 1 (20)$ .

$\frac{- (x) - 1 (20)}{x^{3}}$

Step 1.1.4.5

Factor $- 1$ out of $- (x) - 1 (20)$ .

$\frac{- (x + 20)}{x^{3}}$

Step 1.1.4.6

Rewrite $- (x + 20)$ as $- 1 (x + 20)$ .

$\frac{- 1 (x + 20)}{x^{3}}$

Step 1.1.4.7

Move the negative in front of the fraction.

$f' (x) = - \frac{x + 20}{x^{3}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{x + 20}{x^{3}}$ .

$- \frac{x + 20}{x^{3}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{x + 20}{x^{3}} = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$- \frac{x + 20}{x^{3}} = 0$

Step 2.2

Set the numerator equal to zero.

$x + 20 = 0$

Step 2.3

Subtract $20$ from both sides of the equation.

$x = - 20$

Step 3

The values which make the derivative equal to $0$ are $- 20$ .

$- 20$

Step 4

Find where the derivative is undefined.

Tap for more steps...

Step 4.1

Set the denominator in $\frac{x + 20}{x^{3}}$ equal to $0$ to find where the expression is undefined.

$x^{3} = 0$

Step 4.2

Solve for $x$ .

Tap for more steps...

Step 4.2.1

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

$x = \sqrt[3]{0}$

Step 4.2.2

Simplify $\sqrt[3]{0}$ .

Tap for more steps...

Step 4.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 4.2.2.2

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 5

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

$(- \infty, - 20) \cup (- 20, 0) \cup (0, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 20)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $- 21$ in the expression.

$f' (- 21) = - \frac{(- 21) + 20}{{(- 21)}^{3}}$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Simplify the expression.

Tap for more steps...

Step 6.2.1.1

Add $- 21$ and $20$ .

$f' (- 21) = - \frac{- 1}{{(- 21)}^{3}}$

Step 6.2.1.2

Raise $- 21$ to the power of $3$ .

$f' (- 21) = - \frac{- 1}{- 9261}$

Step 6.2.2

Dividing two negative values results in a positive value.

$f' (- 21) = - \frac{1}{9261}$

Step 6.2.3

The final answer is $- \frac{1}{9261}$ .

$- \frac{1}{9261}$

Step 6.3

At $x = - 21$ the derivative is $- \frac{1}{9261}$ . Since this is negative, the function is decreasing on $(- \infty, - 20)$ .

Decreasing on $(- \infty, - 20)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(- 20, 0)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 7.1

Replace the variable $x$ with $- 10$ in the expression.

$f' (- 10) = - \frac{(- 10) + 20}{{(- 10)}^{3}}$

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Simplify the expression.

Tap for more steps...

Step 7.2.1.1

Add $- 10$ and $20$ .

$f' (- 10) = - \frac{10}{{(- 10)}^{3}}$

Step 7.2.1.2

Raise $- 10$ to the power of $3$ .

$f' (- 10) = - \frac{10}{- 1000}$

Step 7.2.2

Cancel the common factor of $10$ and $- 1000$ .

Tap for more steps...

Step 7.2.2.1

Factor $10$ out of $10$ .

$f' (- 10) = - \frac{10 (1)}{- 1000}$

Step 7.2.2.2

Cancel the common factors.

Tap for more steps...

Step 7.2.2.2.1

Factor $10$ out of $- 1000$ .

$f' (- 10) = - \frac{10 \cdot 1}{10 \cdot - 100}$

Step 7.2.2.2.2

Cancel the common factor.

$f' (- 10) = - \frac{10 \cdot 1}{10 \cdot - 100}$

Step 7.2.2.2.3

Rewrite the expression.

$f' (- 10) = - \frac{1}{- 100}$

Step 7.2.3

Move the negative in front of the fraction.

$f' (- 10) = \frac{1}{100}$

Step 7.2.4

The final answer is $\frac{1}{100}$ .

$\frac{1}{100}$

Step 7.3

At $x = - 10$ the derivative is $\frac{1}{100}$ . Since this is positive, the function is increasing on $(- 20, 0)$ .

Increasing on $(- 20, 0)$ since $f' (x) > 0$

Step 8

Substitute a value from the interval $(0, \infty)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 8.1

Replace the variable $x$ with $1$ in the expression.

$f' (1) = - \frac{(1) + 20}{{(1)}^{3}}$

Step 8.2

Simplify the result.

Tap for more steps...

Step 8.2.1

Add $1$ and $20$ .

$f' (1) = - \frac{21}{1^{3}}$

Step 8.2.2

One to any power is one.

$f' (1) = - \frac{21}{1}$

Step 8.2.3

Divide $21$ by $1$ .

$f' (1) = - 1 \cdot 21$

Step 8.2.4

Multiply $- 1$ by $21$ .

$f' (1) = - 21$

Step 8.2.5

The final answer is $- 21$ .

$- 21$

Step 8.3

At $x = 1$ the derivative is $- 21$ . Since this is negative, the function is decreasing on $(0, \infty)$ .

Decreasing on $(0, \infty)$ since $f' (x) < 0$

Step 9

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

Increasing on: $(- 20, 0)$

Decreasing on: $(- \infty, - 20), (0, \infty)$

Step 10