Calculus Examples

$f (x) = 2 x \ln (x)$

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

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

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

Step 1.1.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \ln (x)$ .

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

Step 1.1.3

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

$2 (x \frac{1}{x} + \ln (x) \frac{d}{d x} [x])$

Step 1.1.4

Differentiate using the Power Rule.

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Step 1.1.4.1

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

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

Step 1.1.4.2

Cancel the common factor of $x$ .

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Step 1.1.4.2.1

Cancel the common factor.

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

Step 1.1.4.2.2

Rewrite the expression.

$2 (1 + \ln (x) \frac{d}{d x} [x])$

Step 1.1.4.3

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

$2 (1 + \ln (x) \cdot 1)$

Step 1.1.4.4

Multiply $\ln (x)$ by $1$ .

$2 (1 + \ln (x))$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Apply the distributive property.

$2 \cdot 1 + 2 \ln (x)$

Step 1.1.5.2

Multiply $2$ by $1$ .

$2 + 2 \ln (x)$

Step 1.1.5.3

Reorder terms.

$f' (x) = 2 \ln (x) + 2$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $2 \ln (x) + 2$ .

$2 \ln (x) + 2$

Step 2

Set the first derivative equal to $0$ then solve the equation $2 \ln (x) + 2 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$2 \ln (x) + 2 = 0$

Step 2.2

Subtract $2$ from both sides of the equation.

$2 \ln (x) = - 2$

Step 2.3

Divide each term in $2 \ln (x) = - 2$ by $2$ and simplify.

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Step 2.3.1

Divide each term in $2 \ln (x) = - 2$ by $2$ .

$\frac{2 \ln (x)}{2} = \frac{- 2}{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{2 \ln (x)}{2} = \frac{- 2}{2}$

Step 2.3.2.1.2

Divide $\ln (x)$ by $1$ .

$\ln (x) = \frac{- 2}{2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $- 2$ by $2$ .

$\ln (x) = - 1$

Step 2.4

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (x)} = e^{- 1}$

Step 2.5

Rewrite $\ln (x) = - 1$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{- 1} = x$

Step 2.6

Solve for $x$ .

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Step 2.6.1

Rewrite the equation as $x = e^{- 1}$ .

$x = e^{- 1}$

Step 2.6.2

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

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

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the argument in $\ln (x)$ less than or equal to $0$ to find where the expression is undefined.

$x \leq 0$

Step 3.2

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 $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 4

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

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Step 4.1

Evaluate at $x = \frac{1}{e}$ .

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Step 4.1.1

Substitute $\frac{1}{e}$ for $x$ .

$2 (\frac{1}{e}) \ln (\frac{1}{e})$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Combine $2$ and $\frac{1}{e}$ .

$\frac{2}{e} \ln (\frac{1}{e})$

Step 4.1.2.2

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

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

Step 4.1.2.3

Rewrite $\ln (1 e^{- 1})$ as $\ln (1) + \ln (e^{- 1})$ .

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

Step 4.1.2.4

Use logarithm rules to move $- 1$ out of the exponent.

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

Step 4.1.2.5

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

$\frac{2}{e} (\ln (1) - 1 \cdot 1)$

Step 4.1.2.6

Multiply $- 1$ by $1$ .

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

Step 4.1.2.7

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

$\frac{2}{e} (0 - 1)$

Step 4.1.2.8

Subtract $1$ from $0$ .

$\frac{2}{e} \cdot - 1$

Step 4.1.2.9

Multiply $\frac{2}{e} \cdot - 1$ .

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Step 4.1.2.9.1

Combine $\frac{2}{e}$ and $- 1$ .

$\frac{2 \cdot - 1}{e}$

Step 4.1.2.9.2

Multiply $2$ by $- 1$ .

$\frac{- 2}{e}$

Step 4.1.2.10

Move the negative in front of the fraction.

$- \frac{2}{e}$

Step 4.2

Evaluate at $x = 0$ .

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Step 4.2.1

Substitute $0$ for $x$ .

$2 (0) \ln (0)$

Step 4.2.2

The natural logarithm of zero is undefined.

Undefined

Step 4.3

List all of the points.

$(\frac{1}{e}, - \frac{2}{e})$

Step 5