Calculus Examples

$f (x) = x - b \ln (x)$

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

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $1 - \frac{b}{x}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{b}{x}]$ .

$f'' (x) = \frac{d}{d x} (1) + \frac{d}{d x} (- \frac{b}{x})$

Step 2.1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$f'' (x) = 0 + \frac{d}{d x} (- \frac{b}{x})$

Step 2.2

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

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Step 2.2.1

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

$f'' (x) = 0 - b \frac{d}{d x} \frac{1}{x}$

Step 2.2.2

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

$f'' (x) = 0 - b \frac{d}{d x} x^{- 1}$

Step 2.2.3

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

$f'' (x) = 0 - b (- x^{- 2})$

Step 2.2.4

Multiply $- 1$ by $- 1$ .

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

Step 2.2.5

Multiply $b$ by $1$ .

$f'' (x) = 0 + b x^{- 2}$

Step 2.3

Simplify.

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Step 2.3.1

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

$f'' (x) = 0 + b (\frac{1}{x^{2}})$

Step 2.3.2

Combine terms.

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Step 2.3.2.1

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

$f'' (x) = 0 + \frac{b}{x^{2}}$

Step 2.3.2.2

Add $0$ and $\frac{b}{x^{2}}$ .

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

Step 3

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

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

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

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

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 $n = 1$ .

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

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

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

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

Step 4.2

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

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

Step 5

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

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Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Subtract $1$ from both sides of the equation.

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

Step 5.3

Find the LCD of the terms in the equation.

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Step 5.3.1

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

$x, 1$

Step 5.3.2

The LCM of one and any expression is the expression.

$x$

Step 5.4

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

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Step 5.4.1

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

$- \frac{b}{x} x = - x$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $x$ .

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Step 5.4.2.1.1

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

$\frac{- b}{x} x = - x$

Step 5.4.2.1.2

Cancel the common factor.

$\frac{- b}{x} x = - x$

Step 5.4.2.1.3

Rewrite the expression.

$- b = - x$

Step 5.5

Solve the equation.

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Step 5.5.1

Rewrite the equation as $- x = - b$ .

$- x = - b$

Step 5.5.2

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

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Step 5.5.2.1

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

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

Step 5.5.2.2

Simplify the left side.

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Step 5.5.2.2.1

Dividing two negative values results in a positive value.

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

Step 5.5.2.2.2

Divide $x$ by $1$ .

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

Step 5.5.2.3

Simplify the right side.

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Step 5.5.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{b}{1}$

Step 5.5.2.3.2

Divide $b$ by $1$ .

$x = b$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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

$x = 0$

Step 6.2

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$b = 0$

Step 7

Critical points to evaluate.

$x = b$

Step 8

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

$\frac{b}{{(b)}^{2}}$

Step 9

Cancel the common factor of $b$ and ${(b)}^{2}$ .

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Step 9.1

Raise $b$ to the power of $1$ .

$\frac{b^{1}}{{(b)}^{2}}$

Step 9.2

Factor $b$ out of $b^{1}$ .

$\frac{b \cdot 1}{{(b)}^{2}}$

Step 9.3

Cancel the common factors.

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Step 9.3.1

Factor $b$ out of ${(b)}^{2}$ .

$\frac{b \cdot 1}{b \cdot b}$

Step 9.3.2

Cancel the common factor.

$\frac{b \cdot 1}{b \cdot b}$

Step 9.3.3

Rewrite the expression.

$\frac{1}{b}$

Step 10

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 11