Calculus Examples

$5 x - 4 \ln (x)$

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 $5 x - 4 \ln (x)$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- 4 \ln (x)]$ .

$\frac{d}{d x} [5 x] + \frac{d}{d x} [- 4 \ln (x)]$

Step 1.1.2

Evaluate $\frac{d}{d x} [5 x]$ .

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Step 1.1.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x] + \frac{d}{d x} [- 4 \ln (x)]$

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 $n = 1$ .

$5 \cdot 1 + \frac{d}{d x} [- 4 \ln (x)]$

Step 1.1.2.3

Multiply $5$ by $1$ .

$5 + \frac{d}{d x} [- 4 \ln (x)]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 4 \ln (x)]$ .

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Step 1.1.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 \ln (x)$ with respect to $x$ is $- 4 \frac{d}{d x} [\ln (x)]$ .

$5 - 4 \frac{d}{d x} [\ln (x)]$

Step 1.1.3.2

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

$5 - 4 \frac{1}{x}$

Step 1.1.3.3

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

$5 + \frac{- 4}{x}$

Step 1.1.3.4

Move the negative in front of the fraction.

$5 - \frac{4}{x}$

Step 1.1.4

Reorder terms.

$f' (x) = - \frac{4}{x} + 5$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{4}{x} + 5$ .

$- \frac{4}{x} + 5$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{4}{x} + 5 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{4}{x} + 5 = 0$

Step 2.2

Subtract $5$ from both sides of the equation.

$- \frac{4}{x} = - 5$

Step 2.3

Find the LCD of the terms in the equation.

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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.

$x, 1$

Step 2.3.2

The LCM of one and any expression is the expression.

$x$

Step 2.4

Multiply each term in $- \frac{4}{x} = - 5$ by $x$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $- \frac{4}{x} = - 5$ by $x$ .

$- \frac{4}{x} x = - 5 x$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $x$ .

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Step 2.4.2.1.1

Move the leading negative in $- \frac{4}{x}$ into the numerator.

$\frac{- 4}{x} x = - 5 x$

Step 2.4.2.1.2

Cancel the common factor.

$\frac{- 4}{x} x = - 5 x$

Step 2.4.2.1.3

Rewrite the expression.

$- 4 = - 5 x$

Step 2.5

Solve the equation.

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Step 2.5.1

Rewrite the equation as $- 5 x = - 4$ .

$- 5 x = - 4$

Step 2.5.2

Divide each term in $- 5 x = - 4$ by $- 5$ and simplify.

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Step 2.5.2.1

Divide each term in $- 5 x = - 4$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- 4}{- 5}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $- 5$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- 4}{- 5}$

Step 2.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 5}$

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{4}{5}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{4}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 4

Evaluate $5 x - 4 \ln (x)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{4}{5}$ .

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Step 4.1.1

Substitute $\frac{4}{5}$ for $x$ .

$5 (\frac{4}{5}) - 4 \ln (\frac{4}{5})$

Step 4.1.2

Simplify each term.

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Step 4.1.2.1

Cancel the common factor of $5$ .

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Step 4.1.2.1.1

Cancel the common factor.

$5 (\frac{4}{5}) - 4 \ln (\frac{4}{5})$

Step 4.1.2.1.2

Rewrite the expression.

$4 - 4 \ln (\frac{4}{5})$

Step 4.1.2.2

Simplify $- 4 \ln (\frac{4}{5})$ by moving $4$ inside the logarithm.

$4 - \ln ({(\frac{4}{5})}^{4})$

Step 4.1.2.3

Apply the product rule to $\frac{4}{5}$ .

$4 - \ln (\frac{4^{4}}{5^{4}})$

Step 4.1.2.4

Raise $4$ to the power of $4$ .

$4 - \ln (\frac{256}{5^{4}})$

Step 4.1.2.5

Raise $5$ to the power of $4$ .

$4 - \ln (\frac{256}{625})$

Step 4.2

Evaluate at $x = 0$ .

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Step 4.2.1

Substitute $0$ for $x$ .

$5 (0) - 4 \ln (0)$

Step 4.2.2

The natural logarithm of zero is undefined.

Undefined

Step 4.3

List all of the points.

$(\frac{4}{5}, 4 - \ln (\frac{256}{625}))$

Step 5