Calculus Examples

$f (x) = x^{2} - 8 \ln (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{2} - 8 \ln (x)$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 \ln (x)]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 \ln (x)]$

Step 1.1.2

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

$2 x + \frac{d}{d x} [- 8 \ln (x)]$

Step 1.2

Evaluate $\frac{d}{d x} [- 8 \ln (x)]$ .

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Step 1.2.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 \ln (x)$ with respect to $x$ is $- 8 \frac{d}{d x} [\ln (x)]$ .

$2 x - 8 \frac{d}{d x} [\ln (x)]$

Step 1.2.2

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

$2 x - 8 \frac{1}{x}$

Step 1.2.3

Combine $- 8$ and $\frac{1}{x}$ .

$2 x + \frac{- 8}{x}$

Step 1.2.4

Move the negative in front of the fraction.

$2 x - \frac{8}{x}$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $2 x - \frac{8}{x}$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{8}{x}]$ .

$f'' (x) = \frac{d}{d x} (2 x) + \frac{d}{d x} (- \frac{8}{x})$

Step 2.2

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 2.2.1

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

$f'' (x) = 2 \frac{d}{d x} (x) + \frac{d}{d x} (- \frac{8}{x})$

Step 2.2.2

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

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

Step 2.2.3

Multiply $2$ by $1$ .

$f'' (x) = 2 + \frac{d}{d x} (- \frac{8}{x})$

Step 2.3

Evaluate $\frac{d}{d x} [- \frac{8}{x}]$ .

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Step 2.3.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- \frac{8}{x}$ with respect to $x$ is $- 8 \frac{d}{d x} [\frac{1}{x}]$ .

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

Step 2.3.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 2.3.3

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

$f'' (x) = 2 - 8 (- x^{- 2})$

Step 2.3.4

Multiply $- 1$ by $- 8$ .

$f'' (x) = 2 + 8 x^{- 2}$

Step 2.4

Simplify.

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Step 2.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = 2 + 8 (\frac{1}{x^{2}})$

Step 2.4.2

Combine $8$ and $\frac{1}{x^{2}}$ .

$f'' (x) = 2 + \frac{8}{x^{2}}$

Step 2.4.3

Reorder terms.

$f'' (x) = \frac{8}{x^{2}} + 2$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$2 x - \frac{8}{x} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Differentiate.

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Step 4.1.1.1

By the Sum Rule, the derivative of $x^{2} - 8 \ln (x)$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 \ln (x)]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 \ln (x)]$

Step 4.1.1.2

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

$2 x + \frac{d}{d x} [- 8 \ln (x)]$

Step 4.1.2

Evaluate $\frac{d}{d x} [- 8 \ln (x)]$ .

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Step 4.1.2.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 \ln (x)$ with respect to $x$ is $- 8 \frac{d}{d x} [\ln (x)]$ .

$2 x - 8 \frac{d}{d x} [\ln (x)]$

Step 4.1.2.2

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

$2 x - 8 \frac{1}{x}$

Step 4.1.2.3

Combine $- 8$ and $\frac{1}{x}$ .

$2 x + \frac{- 8}{x}$

Step 4.1.2.4

Move the negative in front of the fraction.

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

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $2 x - \frac{8}{x}$ .

$2 x - \frac{8}{x}$

Step 5

Set the first derivative equal to $0$ then solve the equation $2 x - \frac{8}{x} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$2 x - \frac{8}{x} = 0$

Step 5.2

Find the LCD of the terms in the equation.

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Step 5.2.1

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

$1, x, 1$

Step 5.2.2

The LCM of one and any expression is the expression.

$x$

Step 5.3

Multiply each term in $2 x - \frac{8}{x} = 0$ by $x$ to eliminate the fractions.

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Step 5.3.1

Multiply each term in $2 x - \frac{8}{x} = 0$ by $x$ .

$2 x \cdot x - \frac{8}{x} x = 0 x$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Simplify each term.

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Step 5.3.2.1.1

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

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Step 5.3.2.1.1.1

Move $x$ .

$2 (x \cdot x) - \frac{8}{x} x = 0 x$

Step 5.3.2.1.1.2

Multiply $x$ by $x$ .

$2 x^{2} - \frac{8}{x} x = 0 x$

Step 5.3.2.1.2

Cancel the common factor of $x$ .

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Step 5.3.2.1.2.1

Move the leading negative in $- \frac{8}{x}$ into the numerator.

$2 x^{2} + \frac{- 8}{x} x = 0 x$

Step 5.3.2.1.2.2

Cancel the common factor.

$2 x^{2} + \frac{- 8}{x} x = 0 x$

Step 5.3.2.1.2.3

Rewrite the expression.

$2 x^{2} - 8 = 0 x$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Multiply $0$ by $x$ .

$2 x^{2} - 8 = 0$

Step 5.4

Solve the equation.

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Step 5.4.1

Add $8$ to both sides of the equation.

$2 x^{2} = 8$

Step 5.4.2

Divide each term in $2 x^{2} = 8$ by $2$ and simplify.

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Step 5.4.2.1

Divide each term in $2 x^{2} = 8$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{8}{2}$

Step 5.4.2.2

Simplify the left side.

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Step 5.4.2.2.1

Cancel the common factor of $2$ .

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Step 5.4.2.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{8}{2}$

Step 5.4.2.2.1.2

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

$x^{2} = \frac{8}{2}$

Step 5.4.2.3

Simplify the right side.

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Step 5.4.2.3.1

Divide $8$ by $2$ .

$x^{2} = 4$

Step 5.4.3

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

$x = \pm \sqrt{4}$

Step 5.4.4

Simplify $\pm \sqrt{4}$ .

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Step 5.4.4.1

Rewrite $4$ as $2^{2}$ .

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

Step 5.4.4.2

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

$x = \pm 2$

Step 5.4.5

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

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Step 5.4.5.1

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

$x = 2$

Step 5.4.5.2

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

$x = - 2$

Step 5.4.5.3

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

$x = 2, - 2$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{8}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 7

Critical points to evaluate.

$x = 2$

Step 8

Evaluate the second derivative at $x = 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{8}{{(2)}^{2}} + 2$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

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

$\frac{8}{4} + 2$

Step 9.1.2

Divide $8$ by $4$ .

$2 + 2$

Step 9.2

Add $2$ and $2$ .

$4$

Step 10

$x = 2$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 2$ is a local minimum

Step 11

Find the y-value when $x = 2$ .

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Step 11.1

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

$f (2) = {(2)}^{2} - 8 \ln (2)$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

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

$f (2) = 4 - 8 \ln (2)$

Step 11.2.1.2

Simplify $- 8 \ln (2)$ by moving $8$ inside the logarithm.

$f (2) = 4 - \ln (2^{8})$

Step 11.2.1.3

Raise $2$ to the power of $8$ .

$f (2) = 4 - \ln (256)$

Step 11.2.2

The final answer is $4 - \ln (256)$ .

$y = 4 - \ln (256)$

Step 12

These are the local extrema for $f (x) = x^{2} - 8 \ln (x)$ .

$(2, 4 - \ln (256))$ is a local minima

Step 13