Calculus Examples

$x^{2} \ln (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

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 $f (x) = x^{2}$ and $g (x) = \ln (x)$ .

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

Step 1.2

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

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

Step 1.3

Differentiate using the Power Rule.

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Step 1.3.1

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

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

Step 1.3.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 1.3.2.1

Factor $x$ out of $x^{2}$ .

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

Step 1.3.2.2

Cancel the common factors.

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Step 1.3.2.2.1

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

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

Step 1.3.2.2.2

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

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

Step 1.3.2.2.3

Cancel the common factor.

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

Step 1.3.2.2.4

Rewrite the expression.

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

Step 1.3.2.2.5

Divide $x$ by $1$ .

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

Step 1.3.3

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

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

Step 1.3.4

Reorder terms.

$2 x \ln (x) + x$

Step 2

Find the second derivative of the function.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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 $g (x) = \ln (x)$ .

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

Step 2.2.3

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

$f'' (x) = 2 (x (\frac{1}{x}) + \ln (x) \frac{d}{d x} (x)) + \frac{d}{d x} (x)$

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

Cancel the common factor of $x$ .

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Step 2.2.6.1

Cancel the common factor.

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

Step 2.2.6.2

Rewrite the expression.

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

Step 2.2.7

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

$f'' (x) = 2 (1 + \ln (x)) + \frac{d}{d x} (x)$

Step 2.3

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

$f'' (x) = 2 (1 + \ln (x)) + 1$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$f'' (x) = 2 \cdot 1 + 2 \ln (x) + 1$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Multiply $2$ by $1$ .

$f'' (x) = 2 + 2 \ln (x) + 1$

Step 2.4.2.2

Add $2$ and $1$ .

$f'' (x) = 2 \ln (x) + 3$

Step 3

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

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

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 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 $f (x) = x^{2}$ and $g (x) = \ln (x)$ .

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

Step 4.1.2

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

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

Step 4.1.3

Differentiate using the Power Rule.

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Step 4.1.3.1

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

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

Step 4.1.3.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 4.1.3.2.1

Factor $x$ out of $x^{2}$ .

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

Step 4.1.3.2.2

Cancel the common factors.

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Step 4.1.3.2.2.1

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

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

Step 4.1.3.2.2.2

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

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

Step 4.1.3.2.2.3

Cancel the common factor.

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

Step 4.1.3.2.2.4

Rewrite the expression.

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

Step 4.1.3.2.2.5

Divide $x$ by $1$ .

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

Step 4.1.3.3

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

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

Step 4.1.3.4

Reorder terms.

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

Step 4.2

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

$2 x \ln (x) + x$

Step 5

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

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Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Subtract $x$ from both sides of the equation.

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Rewrite the expression.

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

Step 5.3.2.2

Cancel the common factor of $x$ .

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

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

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Cancel the common factor of $x$ .

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Step 5.3.3.1.1

Cancel the common factor.

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

Step 5.3.3.1.2

Rewrite the expression.

$\ln (x) = \frac{- 1}{2}$

Step 5.3.3.2

Move the negative in front of the fraction.

$\ln (x) = - \frac{1}{2}$

Step 5.4

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

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

Step 5.5

Rewrite $\ln (x) = - \frac{1}{2}$ 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 $b^{y} = x$ .

$e^{- \frac{1}{2}} = x$

Step 5.6

Solve for $x$ .

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Step 5.6.1

Rewrite the equation as $x = e^{- \frac{1}{2}}$ .

$x = e^{- \frac{1}{2}}$

Step 5.6.2

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

$x = \frac{1}{e^{\frac{1}{2}}}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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

$x \leq 0$

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 $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 7

Critical points to evaluate.

$x = \frac{1}{e^{\frac{1}{2}}}$

Step 8

Evaluate the second derivative at $x = \frac{1}{e^{\frac{1}{2}}}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$2 \ln (\frac{1}{e^{\frac{1}{2}}}) + 3$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

Move $e^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$2 \ln (e^{- \frac{1}{2}}) + 3$

Step 9.1.2

Expand $\ln (e^{- \frac{1}{2}})$ by moving $- \frac{1}{2}$ outside the logarithm.

$2 (- \frac{1}{2} \ln (e)) + 3$

Step 9.1.3

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

$2 (- \frac{1}{2} \cdot 1) + 3$

Step 9.1.4

Multiply $- 1$ by $1$ .

$2 (- \frac{1}{2}) + 3$

Step 9.1.5

Cancel the common factor of $2$ .

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Step 9.1.5.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$2 (\frac{- 1}{2}) + 3$

Step 9.1.5.2

Cancel the common factor.

$2 (\frac{- 1}{2}) + 3$

Step 9.1.5.3

Rewrite the expression.

$- 1 + 3$

Step 9.2

Add $- 1$ and $3$ .

$2$

Step 10

$x = \frac{1}{e^{\frac{1}{2}}}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{1}{e^{\frac{1}{2}}}$ is a local minimum

Step 11

Find the y-value when $x = \frac{1}{e^{\frac{1}{2}}}$ .

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Step 11.1

Replace the variable $x$ with $\frac{1}{e^{\frac{1}{2}}}$ in the expression.

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

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify the expression.

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Step 11.2.1.1

Apply the product rule to $\frac{1}{e^{\frac{1}{2}}}$ .

$f (\frac{1}{e^{\frac{1}{2}}}) = \frac{1^{2}}{{(e^{\frac{1}{2}})}^{2}} \cdot \ln (\frac{1}{e^{\frac{1}{2}}})$

Step 11.2.1.2

One to any power is one.

$f (\frac{1}{e^{\frac{1}{2}}}) = \frac{1}{{(e^{\frac{1}{2}})}^{2}} \cdot \ln (\frac{1}{e^{\frac{1}{2}}})$

Step 11.2.2

Simplify the denominator.

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Step 11.2.2.1

Multiply the exponents in ${(e^{\frac{1}{2}})}^{2}$ .

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Step 11.2.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{1}{e^{\frac{1}{2}}}) = \frac{1}{e^{\frac{1}{2} \cdot 2}} \cdot \ln (\frac{1}{e^{\frac{1}{2}}})$

Step 11.2.2.1.2

Cancel the common factor of $2$ .

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Step 11.2.2.1.2.1

Cancel the common factor.

$f (\frac{1}{e^{\frac{1}{2}}}) = \frac{1}{e^{\frac{1}{2} \cdot 2}} \cdot \ln (\frac{1}{e^{\frac{1}{2}}})$

Step 11.2.2.1.2.2

Rewrite the expression.

$f (\frac{1}{e^{\frac{1}{2}}}) = \frac{1}{e} \cdot \ln (\frac{1}{e^{\frac{1}{2}}})$

Step 11.2.2.2

Simplify.

$f (\frac{1}{e^{\frac{1}{2}}}) = \frac{1}{e} \cdot \ln (\frac{1}{e^{\frac{1}{2}}})$

Step 11.2.3

Move $e^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 11.2.4

Expand $\ln (e^{- \frac{1}{2}})$ by moving $- \frac{1}{2}$ outside the logarithm.

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

Step 11.2.5

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

$f (\frac{1}{e^{\frac{1}{2}}}) = \frac{1}{e} \cdot (- \frac{1}{2} \cdot 1)$

Step 11.2.6

Multiply $- 1$ by $1$ .

$f (\frac{1}{e^{\frac{1}{2}}}) = \frac{1}{e} \cdot (- \frac{1}{2})$

Step 11.2.7

Multiply $\frac{1}{e}$ by $\frac{1}{2}$ .

$f (\frac{1}{e^{\frac{1}{2}}}) = - \frac{1}{e \cdot 2}$

Step 11.2.8

Move $2$ to the left of $e$ .

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

Step 11.2.9

The final answer is $- \frac{1}{2 e}$ .

$y = - \frac{1}{2 e}$

Step 12

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

$(\frac{1}{e^{\frac{1}{2}}}, - \frac{1}{2 e})$ is a local minima

Step 13