Calculus Examples

$f (x) = - 3 x^{2} + 54 \ln (x)$

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

By the Sum Rule, the derivative of $- 3 x^{2} + 54 \ln (x)$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [54 \ln (x)]$ .

$\frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [54 \ln (x)]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- 3 x^{2}]$ .

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Step 1.1.2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{2}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{2}]$ .

$- 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [54 \ln (x)]$

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 $n = 2$ .

$- 3 (2 x) + \frac{d}{d x} [54 \ln (x)]$

Step 1.1.2.3

Multiply $2$ by $- 3$ .

$- 6 x + \frac{d}{d x} [54 \ln (x)]$

Step 1.1.3

Evaluate $\frac{d}{d x} [54 \ln (x)]$ .

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Step 1.1.3.1

Since $54$ is constant with respect to $x$ , the derivative of $54 \ln (x)$ with respect to $x$ is $54 \frac{d}{d x} [\ln (x)]$ .

$- 6 x + 54 \frac{d}{d x} [\ln (x)]$

Step 1.1.3.2

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

$- 6 x + 54 \frac{1}{x}$

Step 1.1.3.3

Combine $54$ and $\frac{1}{x}$ .

$f' (x) = - 6 x + \frac{54}{x}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- 6 x + \frac{54}{x}$ .

$- 6 x + \frac{54}{x}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- 6 x + \frac{54}{x} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- 6 x + \frac{54}{x} = 0$

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.

$1, x, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$x$

Step 2.3

Multiply each term in $- 6 x + \frac{54}{x} = 0$ by $x$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $- 6 x + \frac{54}{x} = 0$ by $x$ .

$- 6 x \cdot x + \frac{54}{x} x = 0 x$

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 $x$ .

$- 6 (x \cdot x) + \frac{54}{x} x = 0 x$

Step 2.3.2.1.1.2

Multiply $x$ by $x$ .

$- 6 x^{2} + \frac{54}{x} x = 0 x$

Step 2.3.2.1.2

Cancel the common factor of $x$ .

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Step 2.3.2.1.2.1

Cancel the common factor.

$- 6 x^{2} + \frac{54}{x} x = 0 x$

Step 2.3.2.1.2.2

Rewrite the expression.

$- 6 x^{2} + 54 = 0 x$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $0$ by $x$ .

$- 6 x^{2} + 54 = 0$

Step 2.4

Solve the equation.

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Step 2.4.1

Subtract $54$ from both sides of the equation.

$- 6 x^{2} = - 54$

Step 2.4.2

Divide each term in $- 6 x^{2} = - 54$ by $- 6$ and simplify.

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Step 2.4.2.1

Divide each term in $- 6 x^{2} = - 54$ by $- 6$ .

$\frac{- 6 x^{2}}{- 6} = \frac{- 54}{- 6}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $- 6$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{- 6 x^{2}}{- 6} = \frac{- 54}{- 6}$

Step 2.4.2.2.1.2

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

$x^{2} = \frac{- 54}{- 6}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Divide $- 54$ by $- 6$ .

$x^{2} = 9$

Step 2.4.3

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

$x = \pm \sqrt{9}$

Step 2.4.4

Simplify $\pm \sqrt{9}$ .

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Step 2.4.4.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

Step 2.4.4.2

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

$x = \pm 3$

Step 2.4.5

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

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Step 2.4.5.1

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

$x = 3$

Step 2.4.5.2

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

$x = - 3$

Step 2.4.5.3

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

$x = 3, - 3$

Step 3

The values which make the derivative equal to $0$ are $3, - 3$ .

$3, - 3$

Step 4

Set the denominator in $\frac{54}{x}$ 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.

$(- \infty, - 3) \cup (- 3, 0) \cup (0, 3) \cup (3, \infty)$

Step 6

Exclude the intervals that are not in the domain.

Step 7

Substitute a value from the interval $(- 3, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- \frac{3}{2}$ in the expression.

$f' (- \frac{3}{2}) = - 6 (- \frac{3}{2}) + \frac{54}{- \frac{3}{2}}$

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

Cancel the common factor of $2$ .

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Step 7.2.1.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$f' (- \frac{3}{2}) = - 6 (\frac{- 3}{2}) + \frac{54}{- \frac{3}{2}}$

Step 7.2.1.1.2

Factor $2$ out of $- 6$ .

$f' (- \frac{3}{2}) = 2 (- 3) (\frac{- 3}{2}) + \frac{54}{- \frac{3}{2}}$

Step 7.2.1.1.3

Cancel the common factor.

$f' (- \frac{3}{2}) = 2 \cdot (- 3 (\frac{- 3}{2})) + \frac{54}{- \frac{3}{2}}$

Step 7.2.1.1.4

Rewrite the expression.

$f' (- \frac{3}{2}) = - 3 \cdot - 3 + \frac{54}{- \frac{3}{2}}$

Step 7.2.1.2

Multiply $- 3$ by $- 3$ .

$f' (- \frac{3}{2}) = 9 + \frac{54}{- \frac{3}{2}}$

Step 7.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$f' (- \frac{3}{2}) = 9 + 54 (- \frac{2}{3})$

Step 7.2.1.4

Cancel the common factor of $3$ .

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Step 7.2.1.4.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$f' (- \frac{3}{2}) = 9 + 54 (\frac{- 2}{3})$

Step 7.2.1.4.2

Factor $3$ out of $54$ .

$f' (- \frac{3}{2}) = 9 + 3 (18) (\frac{- 2}{3})$

Step 7.2.1.4.3

Cancel the common factor.

$f' (- \frac{3}{2}) = 9 + 3 \cdot (18 (\frac{- 2}{3}))$

Step 7.2.1.4.4

Rewrite the expression.

$f' (- \frac{3}{2}) = 9 + 18 \cdot - 2$

Step 7.2.1.5

Multiply $18$ by $- 2$ .

$f' (- \frac{3}{2}) = 9 - 36$

Step 7.2.2

Subtract $36$ from $9$ .

$f' (- \frac{3}{2}) = - 27$

Step 7.2.3

The final answer is $- 27$ .

$- 27$

Step 8

Exclude the intervals that are not in the domain.

$(0, 3)$

Step 9

Substitute a value from the interval $(0, 3)$ into the derivative to determine if the function is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $\frac{3}{2}$ in the expression.

$f' (\frac{3}{2}) = - 6 (\frac{3}{2}) + \frac{54}{\frac{3}{2}}$

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

Factor $2$ out of $- 6$ .

$f' (\frac{3}{2}) = 2 (- 3) (\frac{3}{2}) + \frac{54}{\frac{3}{2}}$

Step 9.2.1.1.2

Cancel the common factor.

$f' (\frac{3}{2}) = 2 \cdot (- 3 (\frac{3}{2})) + \frac{54}{\frac{3}{2}}$

Step 9.2.1.1.3

Rewrite the expression.

$f' (\frac{3}{2}) = - 3 \cdot 3 + \frac{54}{\frac{3}{2}}$

Step 9.2.1.2

Multiply $- 3$ by $3$ .

$f' (\frac{3}{2}) = - 9 + \frac{54}{\frac{3}{2}}$

Step 9.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$f' (\frac{3}{2}) = - 9 + 54 (\frac{2}{3})$

Step 9.2.1.4

Cancel the common factor of $3$ .

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Step 9.2.1.4.1

Factor $3$ out of $54$ .

$f' (\frac{3}{2}) = - 9 + 3 (18) (\frac{2}{3})$

Step 9.2.1.4.2

Cancel the common factor.

$f' (\frac{3}{2}) = - 9 + 3 \cdot (18 (\frac{2}{3}))$

Step 9.2.1.4.3

Rewrite the expression.

$f' (\frac{3}{2}) = - 9 + 18 \cdot 2$

Step 9.2.1.5

Multiply $18$ by $2$ .

$f' (\frac{3}{2}) = - 9 + 36$

Step 9.2.2

Add $- 9$ and $36$ .

$f' (\frac{3}{2}) = 27$

Step 9.2.3

The final answer is $27$ .

$27$

Step 9.3

At $x = \frac{3}{2}$ the derivative is $27$ . Since this is positive, the function is increasing on $(0, 3)$ .

Increasing on $(0, 3)$ since $f' (x) > 0$

Step 10

Exclude the intervals that are not in the domain.

$(0, 3), (3, \infty)$

Step 11

Substitute a value from the interval $(3, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 11.1

Replace the variable $x$ with $4$ in the expression.

$f' (4) = - 6 \cdot 4 + \frac{54}{4}$

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

Multiply $- 6$ by $4$ .

$f' (4) = - 24 + \frac{54}{4}$

Step 11.2.1.2

Cancel the common factor of $54$ and $4$ .

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Step 11.2.1.2.1

Factor $2$ out of $54$ .

$f' (4) = - 24 + \frac{2 (27)}{4}$

Step 11.2.1.2.2

Cancel the common factors.

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Step 11.2.1.2.2.1

Factor $2$ out of $4$ .

$f' (4) = - 24 + \frac{2 \cdot 27}{2 \cdot 2}$

Step 11.2.1.2.2.2

Cancel the common factor.

$f' (4) = - 24 + \frac{2 \cdot 27}{2 \cdot 2}$

Step 11.2.1.2.2.3

Rewrite the expression.

$f' (4) = - 24 + \frac{27}{2}$

Step 11.2.2

To write $- 24$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (4) = - 24 \cdot \frac{2}{2} + \frac{27}{2}$

Step 11.2.3

Combine $- 24$ and $\frac{2}{2}$ .

$f' (4) = \frac{- 24 \cdot 2}{2} + \frac{27}{2}$

Step 11.2.4

Combine the numerators over the common denominator.

$f' (4) = \frac{- 24 \cdot 2 + 27}{2}$

Step 11.2.5

Simplify the numerator.

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Step 11.2.5.1

Multiply $- 24$ by $2$ .

$f' (4) = \frac{- 48 + 27}{2}$

Step 11.2.5.2

Add $- 48$ and $27$ .

$f' (4) = \frac{- 21}{2}$

Step 11.2.6

Move the negative in front of the fraction.

$f' (4) = - \frac{21}{2}$

Step 11.2.7

The final answer is $- \frac{21}{2}$ .

$- \frac{21}{2}$

Step 11.3

At $x = 4$ the derivative is $- \frac{21}{2}$ . Since this is negative, the function is decreasing on $(3, \infty)$ .

Decreasing on $(3, \infty)$ since $f' (x) < 0$

Step 12

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

Increasing on: $(0, 3)$

Decreasing on: $(3, \infty)$

Step 13