Calculus Examples

$f (x) = x^{2} - x - \ln (x)$

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.

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Step 1.1.1.1

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

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

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 $n = 2$ .

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Multiply $- 1$ by $1$ .

$2 x - 1 + \frac{d}{d x} [- \ln (x)]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- \ln (x)]$ .

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Step 1.1.3.1

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

$2 x - 1 - \frac{d}{d x} [\ln (x)]$

Step 1.1.3.2

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

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

Step 1.1.4

Reorder terms.

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

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

$1, x, 1, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$x$

Step 2.3

Multiply each term in $2 x - \frac{1}{x} - 1 = 0$ by $x$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $2 x - \frac{1}{x} - 1 = 0$ by $x$ .

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.3.2.1.1.1

Move $x$ .

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

Step 2.3.2.1.1.2

Multiply $x$ by $x$ .

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

Step 2.3.2.1.2

Cancel the common factor of $x$ .

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Step 2.3.2.1.2.1

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

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

Step 2.3.2.1.2.2

Cancel the common factor.

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

Step 2.3.2.1.2.3

Rewrite the expression.

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

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $0$ by $x$ .

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

Step 2.4

Solve the equation.

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Step 2.4.1

Factor by grouping.

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Step 2.4.1.1

Reorder terms.

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

Step 2.4.1.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = - 1$ .

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Step 2.4.1.2.1

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

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

Step 2.4.1.2.2

Rewrite $- 1$ as $1$ plus $- 2$

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

Step 2.4.1.2.3

Apply the distributive property.

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

Step 2.4.1.2.4

Multiply $x$ by $1$ .

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

Step 2.4.1.3

Factor out the greatest common factor from each group.

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Step 2.4.1.3.1

Group the first two terms and the last two terms.

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

Step 2.4.1.3.2

Factor out the greatest common factor (GCF) from each group.

$x (2 x + 1) - (2 x + 1) = 0$

Step 2.4.1.4

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$(2 x + 1) (x - 1) = 0$

Step 2.4.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x + 1 = 0$

$x - 1 = 0$

Step 2.4.3

Set $2 x + 1$ equal to $0$ and solve for $x$ .

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Step 2.4.3.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 2.4.3.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 2.4.3.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 2.4.3.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 2.4.3.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

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

Step 2.4.3.2.2.2

Simplify the left side.

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Step 2.4.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.4.3.2.2.2.1.1

Cancel the common factor.

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

Step 2.4.3.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.4.3.2.2.3

Simplify the right side.

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Step 2.4.3.2.2.3.1

Move the negative in front of the fraction.

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

Step 2.4.4

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 2.4.4.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 2.4.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.4.5

The final solution is all the values that make $(2 x + 1) (x - 1) = 0$ true.

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

Step 3

The values which make the derivative equal to $0$ are $- \frac{1}{2}, 1$ .

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

Step 4

Set the denominator in $\frac{1}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 5

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

$(- \infty, - \frac{1}{2}) \cup (- \frac{1}{2}, 0) \cup (0, 1) \cup (1, \infty)$

Step 6

Exclude the intervals that are not in the domain.

Step 7

Substitute a value from the interval $(- 0.5, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 0.25$ in the expression.

$f' (- 0.25) = 2 (- 0.25) - \frac{1}{- 0.25} - 1$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Multiply $2$ by $- 0.25$ .

$f' (- 0.25) = - 0.5 - \frac{1}{- 0.25} - 1$

Step 7.2.1.2

Divide $1$ by $- 0.25$ .

$f' (- 0.25) = - 0.5 + 4 - 1$

Step 7.2.1.3

Multiply $- 1$ by $- 4$ .

$f' (- 0.25) = - 0.5 + 4 - 1$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Add $- 0.5$ and $4$ .

$f' (- 0.25) = 3.5 - 1$

Step 7.2.2.2

Subtract $1$ from $3.5$ .

$f' (- 0.25) = 2.5$

Step 7.2.3

The final answer is $2.5$ .

$2.5$

Step 8

Exclude the intervals that are not in the domain.

$(0, 1)$

Step 9

Substitute a value from the interval $(0, 1)$ into the derivative to determine if the function is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

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

Step 9.2

Simplify the result.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

Cancel the common factor of $2$ .

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Step 9.2.1.1.1

Cancel the common factor.

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

Step 9.2.1.1.2

Rewrite the expression.

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

Step 9.2.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 9.2.1.3

Multiply $- (1 \cdot 2)$ .

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Step 9.2.1.3.1

Multiply $2$ by $1$ .

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

Step 9.2.1.3.2

Multiply $- 1$ by $2$ .

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

Step 9.2.2

Simplify by subtracting numbers.

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Step 9.2.2.1

Subtract $2$ from $1$ .

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

Step 9.2.2.2

Subtract $1$ from $- 1$ .

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

Step 9.2.3

The final answer is $- 2$ .

$- 2$

Step 9.3

At $x = \frac{1}{2}$ the derivative is $- 2$ . Since this is negative, the function is decreasing on $(0, 1)$ .

Decreasing on $(0, 1)$ since $f' (x) < 0$

Step 10

Exclude the intervals that are not in the domain.

$(0, 1), (1, \infty)$

Step 11

Substitute a value from the interval $(1, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 11.1

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

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

Step 11.2

Simplify the result.

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Step 11.2.1

Multiply $2$ by $2$ .

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

Step 11.2.2

Find the common denominator.

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Step 11.2.2.1

Write $4$ as a fraction with denominator $1$ .

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

Step 11.2.2.2

Multiply $\frac{4}{1}$ by $\frac{2}{2}$ .

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

Step 11.2.2.3

Multiply $\frac{4}{1}$ by $\frac{2}{2}$ .

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

Step 11.2.2.4

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

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

Step 11.2.2.5

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

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

Step 11.2.2.6

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

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

Step 11.2.3

Combine the numerators over the common denominator.

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

Step 11.2.4

Simplify each term.

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Step 11.2.4.1

Multiply $4$ by $2$ .

$f' (2) = \frac{8 - 1 - 1 \cdot 2}{2}$

Step 11.2.4.2

Multiply $- 1$ by $2$ .

$f' (2) = \frac{8 - 1 - 2}{2}$

Step 11.2.5

Simplify by subtracting numbers.

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Step 11.2.5.1

Subtract $1$ from $8$ .

$f' (2) = \frac{7 - 2}{2}$

Step 11.2.5.2

Subtract $2$ from $7$ .

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

Step 11.2.6

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

$\frac{5}{2}$

Step 11.3

At $x = 2$ the derivative is $\frac{5}{2}$ . Since this is positive, the function is increasing on $(1, \infty)$ .

Increasing on $(1, \infty)$ since $f' (x) > 0$

Step 12

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

Increasing on: $(1, \infty)$

Decreasing on: $(0, 1)$

Step 13