Calculus Examples

$f (x) = x - 4 \sqrt{x}$

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.

Tap for more steps...

Step 1.1.1.1

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

$f' (x) = \frac{d}{d x} (x) + \frac{d}{d x} (- 4 \sqrt{x})$

Step 1.1.1.2

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

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

Step 1.1.2

Evaluate $\frac{d}{d x} [- 4 \sqrt{x}]$ .

Tap for more steps...

Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

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 $n = \frac{1}{2}$ .

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

Step 1.1.2.4

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

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

Step 1.1.2.5

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

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

Step 1.1.2.6

Combine the numerators over the common denominator.

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

Step 1.1.2.7

Simplify the numerator.

Tap for more steps...

Step 1.1.2.7.1

Multiply $- 1$ by $2$ .

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

Step 1.1.2.7.2

Subtract $2$ from $1$ .

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

Step 1.1.2.8

Move the negative in front of the fraction.

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

Step 1.1.2.9

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

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

Step 1.1.2.10

Combine $- 4$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

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

Step 1.1.2.11

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

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

Step 1.1.2.12

Factor $2$ out of $- 4$ .

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

Step 1.1.2.13

Cancel the common factors.

Tap for more steps...

Step 1.1.2.13.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

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

Step 1.1.2.13.2

Cancel the common factor.

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

Step 1.1.2.13.3

Rewrite the expression.

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

Step 1.1.2.14

Move the negative in front of the fraction.

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

Step 1.2

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

$1 - \frac{2}{x^{\frac{1}{2}}}$

Step 2

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

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Subtract $1$ from both sides of the equation.

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

Step 2.3

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{\frac{1}{2}}, 1$

Step 2.3.2

The LCM of one and any expression is the expression.

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

Step 2.4

Multiply each term in $- \frac{2}{x^{\frac{1}{2}}} = - 1$ by $x^{\frac{1}{2}}$ to eliminate the fractions.

Tap for more steps...

Step 2.4.1

Multiply each term in $- \frac{2}{x^{\frac{1}{2}}} = - 1$ by $x^{\frac{1}{2}}$ .

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

Step 2.4.2

Simplify the left side.

Tap for more steps...

Step 2.4.2.1

Cancel the common factor of $x^{\frac{1}{2}}$ .

Tap for more steps...

Step 2.4.2.1.1

Move the leading negative in $- \frac{2}{x^{\frac{1}{2}}}$ into the numerator.

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

Step 2.4.2.1.2

Cancel the common factor.

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

Step 2.4.2.1.3

Rewrite the expression.

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

Step 2.5

Solve the equation.

Tap for more steps...

Step 2.5.1

Rewrite the equation as $- x^{\frac{1}{2}} = - 2$ .

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

Step 2.5.2

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

Tap for more steps...

Step 2.5.2.1

Divide each term in $- x^{\frac{1}{2}} = - 2$ by $- 1$ .

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

Step 2.5.2.2

Simplify the left side.

Tap for more steps...

Step 2.5.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.5.2.2.2

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

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

Step 2.5.2.3

Simplify the right side.

Tap for more steps...

Step 2.5.2.3.1

Divide $- 2$ by $- 1$ .

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

Step 2.5.3

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 2.5.4

Simplify the exponent.

Tap for more steps...

Step 2.5.4.1

Simplify the left side.

Tap for more steps...

Step 2.5.4.1.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.5.4.1.1.1

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

Tap for more steps...

Step 2.5.4.1.1.1.1

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

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

Step 2.5.4.1.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.5.4.1.1.1.2.1

Cancel the common factor.

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

Step 2.5.4.1.1.1.2.2

Rewrite the expression.

$x^{1} = 2^{2}$

Step 2.5.4.1.1.2

Simplify.

$x = 2^{2}$

Step 2.5.4.2

Simplify the right side.

Tap for more steps...

Step 2.5.4.2.1

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

$x = 4$

Step 3

The values which make the derivative equal to $0$ are $4$ .

$4$

Step 4

Find where the derivative is undefined.

Tap for more steps...

Step 4.1

Convert expressions with fractional exponents to radicals.

Tap for more steps...

Step 4.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$1 - \frac{2}{\sqrt{x^{1}}}$

Step 4.1.2

Anything raised to $1$ is the base itself.

$1 - \frac{2}{\sqrt{x}}$

Step 4.2

Set the denominator in $\frac{2}{\sqrt{x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{x} = 0$

Step 4.3

Solve for $x$ .

Tap for more steps...

Step 4.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x}}^{2} = 0^{2}$

Step 4.3.2

Simplify each side of the equation.

Tap for more steps...

Step 4.3.2.1

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

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

Step 4.3.2.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.2.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 4.3.2.2.1.1

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

Tap for more steps...

Step 4.3.2.2.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.3.2.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.3.2.2.1.1.2.2

Rewrite the expression.

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

Step 4.3.2.2.1.2

Simplify.

$x = 0^{2}$

Step 4.3.2.3

Simplify the right side.

Tap for more steps...

Step 4.3.2.3.1

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

$x = 0$

Step 4.4

Set the radicand in $\sqrt{x}$ less than $0$ to find where the expression is undefined.

$x < 0$

Step 4.5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 5

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

$(- \infty, 0) \cup (0, 4) \cup (4, \infty)$

Step 6

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

Tap for more steps...

Step 6.1

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

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

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Simplify the denominator.

Tap for more steps...

Step 6.2.1.1.1

Rewrite ${(- 1)}^{\frac{1}{2}}$ as $\sqrt{{(- 1)}^{1}}$ .

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

Step 6.2.1.1.2

Evaluate the exponent.

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

Step 6.2.1.1.3

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 6.2.1.2

Multiply the numerator and denominator of $\frac{2}{i}$ by the conjugate of $i$ to make the denominator real.

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

Step 6.2.1.3

Multiply.

Tap for more steps...

Step 6.2.1.3.1

Combine.

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

Step 6.2.1.3.2

Simplify the denominator.

Tap for more steps...

Step 6.2.1.3.2.1

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

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

Step 6.2.1.3.2.2

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

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

Step 6.2.1.3.2.3

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

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

Step 6.2.1.3.2.4

Add $1$ and $1$ .

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

Step 6.2.1.3.2.5

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

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

Step 6.2.1.4

Move the negative one from the denominator of $\frac{2 i}{- 1}$ .

$f' (- 1) = 1 - (- 1 \cdot (2 i))$

Step 6.2.1.5

Rewrite $- 1 \cdot (2 i)$ as $- (2 i)$ .

$f' (- 1) = 1 + 2 i$

Step 6.2.1.6

Multiply $2$ by $- 1$ .

$f' (- 1) = 1 - (- 2 i)$

Step 6.2.1.7

Multiply $- 2$ by $- 1$ .

$f' (- 1) = 1 + 2 i$

Step 6.2.2

The final answer is $1 + 2 i$ .

$1 + 2 i$

Step 6.3

At $x = - 1$ the derivative is $1 + 2 i$ . Since this contains an imaginary number, the function does not exist on $(- \infty, 0)$ .

Function is not real on $(- \infty, 0)$ since $f' (x)$ is imaginary

Step 7

Substitute a value from the interval $(0, 4)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 7.1

Replace the variable $x$ with $2$ in the expression.

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

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Simplify each term.

Tap for more steps...

Step 7.2.1.1

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

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

Step 7.2.1.2

Multiply $2$ by $2^{- \frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 7.2.1.2.1

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

Tap for more steps...

Step 7.2.1.2.1.1

Raise $2$ to the power of $1$ .

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

Step 7.2.1.2.1.2

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

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

Step 7.2.1.2.2

Write $1$ as a fraction with a common denominator.

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

Step 7.2.1.2.3

Combine the numerators over the common denominator.

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

Step 7.2.1.2.4

Subtract $1$ from $2$ .

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

Step 7.2.2

The final answer is $1 - 2^{\frac{1}{2}}$ .

$1 - 2^{\frac{1}{2}}$

Step 7.3

Simplify.

$- 0.41421356$

Step 7.4

At $x = 2$ the derivative is $- 0.41421356$ . Since this is negative, the function is decreasing on $(0, 4)$ .

Decreasing on $(0, 4)$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(4, \infty)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 8.1

Replace the variable $x$ with $5$ in the expression.

$f' (5) = 1 - \frac{2}{{(5)}^{\frac{1}{2}}}$

Step 8.2

Simplify the result.

Tap for more steps...

Step 8.2.1

Remove parentheses.

$f' (5) = 1 - \frac{2}{5^{\frac{1}{2}}}$

Step 8.2.2

The final answer is $1 - \frac{2}{5^{\frac{1}{2}}}$ .

$1 - \frac{2}{5^{\frac{1}{2}}}$

Step 8.3

Simplify.

$0.1055728$

Step 8.4

At $x = 5$ the derivative is $0.1055728$ . Since this is positive, the function is increasing on $(4, \infty)$ .

Increasing on $(4, \infty)$ since $f' (x) > 0$

Step 9

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

Increasing on: $(4, \infty)$

Decreasing on: $(0, 4)$

Step 10