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

Find the values where the derivative is undefined.

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Step 3.1

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

$x = 0$

Step 4

Evaluate $x^{2} - x - \ln (x)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - \frac{1}{2}$ .

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Step 4.1.1

Substitute $- \frac{1}{2}$ for $x$ .

${(- \frac{1}{2})}^{2} - (- \frac{1}{2}) - \ln (- \frac{1}{2})$

Step 4.1.2

The natural logarithm of a negative number is undefined.

Undefined

Step 4.2

Evaluate at $x = 1$ .

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Step 4.2.1

Substitute $1$ for $x$ .

${(1)}^{2} - (1) - \ln (1)$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

One to any power is one.

$1 - (1) - \ln (1)$

Step 4.2.2.1.2

Multiply $- 1$ by $1$ .

$1 - 1 - \ln (1)$

Step 4.2.2.1.3

The natural logarithm of $1$ is $0$ .

$1 - 1 - 0$

Step 4.2.2.1.4

Multiply $- 1$ by $0$ .

$1 - 1 + 0$

Step 4.2.2.2

Simplify by adding and subtracting.

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Step 4.2.2.2.1

Subtract $1$ from $1$ .

$0 + 0$

Step 4.2.2.2.2

Add $0$ and $0$ .

$0$

Step 4.3

Evaluate at $x = 0$ .

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Step 4.3.1

Substitute $0$ for $x$ .

${(0)}^{2} - (0) - \ln (0)$

Step 4.3.2

The natural logarithm of zero is undefined.

Undefined

Step 4.4

List all of the points.

$(1, 0)$

Step 5