Calculus Examples

$y = x^{3} \ln (x)$

Step 1

Write $y = x^{3} \ln (x)$ as a function.

$f (x) = x^{3} \ln (x)$

Step 2

Find the first derivative of the function.

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Step 2.1

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 $f (x) = x^{3}$ and $g (x) = \ln (x)$ .

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

Step 2.2

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

$x^{3} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{3}]$

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

Combine $x^{3}$ and $\frac{1}{x}$ .

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

Step 2.3.2

Cancel the common factor of $x^{3}$ and $x$ .

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Step 2.3.2.1

Factor $x$ out of $x^{3}$ .

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

Step 2.3.2.2

Cancel the common factors.

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Step 2.3.2.2.1

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

$\frac{x \cdot x^{2}}{x^{1}} + \ln (x) \frac{d}{d x} [x^{3}]$

Step 2.3.2.2.2

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

$\frac{x \cdot x^{2}}{x \cdot 1} + \ln (x) \frac{d}{d x} [x^{3}]$

Step 2.3.2.2.3

Cancel the common factor.

$\frac{x \cdot x^{2}}{x \cdot 1} + \ln (x) \frac{d}{d x} [x^{3}]$

Step 2.3.2.2.4

Rewrite the expression.

$\frac{x^{2}}{1} + \ln (x) \frac{d}{d x} [x^{3}]$

Step 2.3.2.2.5

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

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

Step 2.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$x^{2} + \ln (x) (3 x^{2})$

Step 2.3.4

Reorder terms.

$x^{2} + 3 x^{2} \ln (x)$

Step 3

Find the second derivative of the function.

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Step 3.1

Differentiate.

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Step 3.1.1

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

$f'' (x) = \frac{d}{d x} (x^{2}) + \frac{d}{d x} (3 x^{2} \ln (x))$

Step 3.1.2

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

$f'' (x) = 2 x + \frac{d}{d x} (3 x^{2} \ln (x))$

Step 3.2

Evaluate $\frac{d}{d x} [3 x^{2} \ln (x)]$ .

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Step 3.2.1

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

$f'' (x) = 2 x + 3 \frac{d}{d x} (x^{2} \ln (x))$

Step 3.2.2

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

$f'' (x) = 2 x + 3 (x^{2} \frac{d}{d x} (\ln (x)) + \ln (x) \frac{d}{d x} (x^{2}))$

Step 3.2.3

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

$f'' (x) = 2 x + 3 (x^{2} (\frac{1}{x}) + \ln (x) \frac{d}{d x} (x^{2}))$

Step 3.2.4

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

$f'' (x) = 2 x + 3 (x^{2} (\frac{1}{x}) + \ln (x) (2 x))$

Step 3.2.5

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

$f'' (x) = 2 x + 3 (\frac{x^{2}}{x} + \ln (x) (2 x))$

Step 3.2.6

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

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Step 3.2.6.1

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

$f'' (x) = 2 x + 3 (\frac{x \cdot x}{x} + \ln (x) (2 x))$

Step 3.2.6.2

Cancel the common factors.

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Step 3.2.6.2.1

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

$f'' (x) = 2 x + 3 (\frac{x \cdot x}{x} + \ln (x) (2 x))$

Step 3.2.6.2.2

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

$f'' (x) = 2 x + 3 (\frac{x \cdot x}{x \cdot 1} + \ln (x) (2 x))$

Step 3.2.6.2.3

Cancel the common factor.

$f'' (x) = 2 x + 3 (\frac{x \cdot x}{x \cdot 1} + \ln (x) (2 x))$

Step 3.2.6.2.4

Rewrite the expression.

$f'' (x) = 2 x + 3 (\frac{x}{1} + \ln (x) (2 x))$

Step 3.2.6.2.5

Divide $x$ by $1$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Combine terms.

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Step 3.3.2.1

Multiply $2$ by $3$ .

$f'' (x) = 2 x + 3 x + 6 (\ln (x) (x))$

Step 3.3.2.2

Add $2 x$ and $3 x$ .

$f'' (x) = 5 x + 6 \ln (x) x$

Step 3.3.3

Reorder terms.

$f'' (x) = 6 x \ln (x) + 5 x$

Step 4

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

$x^{2} + 3 x^{2} \ln (x) = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

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 $f (x) = x^{3}$ and $g (x) = \ln (x)$ .

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

Step 5.1.2

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

$x^{3} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{3}]$

Step 5.1.3

Differentiate using the Power Rule.

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Step 5.1.3.1

Combine $x^{3}$ and $\frac{1}{x}$ .

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

Step 5.1.3.2

Cancel the common factor of $x^{3}$ and $x$ .

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Step 5.1.3.2.1

Factor $x$ out of $x^{3}$ .

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

Step 5.1.3.2.2

Cancel the common factors.

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Step 5.1.3.2.2.1

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

$\frac{x \cdot x^{2}}{x^{1}} + \ln (x) \frac{d}{d x} [x^{3}]$

Step 5.1.3.2.2.2

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

$\frac{x \cdot x^{2}}{x \cdot 1} + \ln (x) \frac{d}{d x} [x^{3}]$

Step 5.1.3.2.2.3

Cancel the common factor.

$\frac{x \cdot x^{2}}{x \cdot 1} + \ln (x) \frac{d}{d x} [x^{3}]$

Step 5.1.3.2.2.4

Rewrite the expression.

$\frac{x^{2}}{1} + \ln (x) \frac{d}{d x} [x^{3}]$

Step 5.1.3.2.2.5

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

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

Step 5.1.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$x^{2} + \ln (x) (3 x^{2})$

Step 5.1.3.4

Reorder terms.

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

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $x^{2} + 3 x^{2} \ln (x)$ .

$x^{2} + 3 x^{2} \ln (x)$

Step 6

Set the first derivative equal to $0$ then solve the equation $x^{2} + 3 x^{2} \ln (x) = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$x^{2} + 3 x^{2} \ln (x) = 0$

Step 6.2

Subtract $x^{2}$ from both sides of the equation.

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

Step 6.3

Divide each term in $3 x^{2} \ln (x) = - x^{2}$ by $3 x^{2}$ and simplify.

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Step 6.3.1

Divide each term in $3 x^{2} \ln (x) = - x^{2}$ by $3 x^{2}$ .

$\frac{3 x^{2} \ln (x)}{3 x^{2}} = \frac{- x^{2}}{3 x^{2}}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $3$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{3 x^{2} \ln (x)}{3 x^{2}} = \frac{- x^{2}}{3 x^{2}}$

Step 6.3.2.1.2

Rewrite the expression.

$\frac{x^{2} \ln (x)}{x^{2}} = \frac{- x^{2}}{3 x^{2}}$

Step 6.3.2.2

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

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Step 6.3.2.2.1

Cancel the common factor.

$\frac{x^{2} \ln (x)}{x^{2}} = \frac{- x^{2}}{3 x^{2}}$

Step 6.3.2.2.2

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

$\ln (x) = \frac{- x^{2}}{3 x^{2}}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

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

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Step 6.3.3.1.1

Cancel the common factor.

$\ln (x) = \frac{- x^{2}}{3 x^{2}}$

Step 6.3.3.1.2

Rewrite the expression.

$\ln (x) = \frac{- 1}{3}$

Step 6.3.3.2

Move the negative in front of the fraction.

$\ln (x) = - \frac{1}{3}$

Step 6.4

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

$e^{\ln (x)} = e^{- \frac{1}{3}}$

Step 6.5

Rewrite $\ln (x) = - \frac{1}{3}$ 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 $b^{y} = x$ .

$e^{- \frac{1}{3}} = x$

Step 6.6

Solve for $x$ .

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Step 6.6.1

Rewrite the equation as $x = e^{- \frac{1}{3}}$ .

$x = e^{- \frac{1}{3}}$

Step 6.6.2

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

$x = \frac{1}{e^{\frac{1}{3}}}$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

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

$x \leq 0$

Step 7.2

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 8

Critical points to evaluate.

$x = \frac{1}{e^{\frac{1}{3}}}$

Step 9

Evaluate the second derivative at $x = \frac{1}{e^{\frac{1}{3}}}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$6 (\frac{1}{e^{\frac{1}{3}}}) \ln (\frac{1}{e^{\frac{1}{3}}}) + 5 (\frac{1}{e^{\frac{1}{3}}})$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify each term.

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Step 10.1.1

Combine $6$ and $\frac{1}{e^{\frac{1}{3}}}$ .

$\frac{6}{e^{\frac{1}{3}}} \ln (\frac{1}{e^{\frac{1}{3}}}) + 5 (\frac{1}{e^{\frac{1}{3}}})$

Step 10.1.2

Move $e^{\frac{1}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{6}{e^{\frac{1}{3}}} \ln (e^{- \frac{1}{3}}) + 5 (\frac{1}{e^{\frac{1}{3}}})$

Step 10.1.3

Expand $\ln (e^{- \frac{1}{3}})$ by moving $- \frac{1}{3}$ outside the logarithm.

$\frac{6}{e^{\frac{1}{3}}} (- \frac{1}{3} \ln (e)) + 5 (\frac{1}{e^{\frac{1}{3}}})$

Step 10.1.4

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

$\frac{6}{e^{\frac{1}{3}}} (- \frac{1}{3} \cdot 1) + 5 (\frac{1}{e^{\frac{1}{3}}})$

Step 10.1.5

Multiply $- 1$ by $1$ .

$\frac{6}{e^{\frac{1}{3}}} (- \frac{1}{3}) + 5 (\frac{1}{e^{\frac{1}{3}}})$

Step 10.1.6

Cancel the common factor of $3$ .

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Step 10.1.6.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$\frac{6}{e^{\frac{1}{3}}} \cdot \frac{- 1}{3} + 5 (\frac{1}{e^{\frac{1}{3}}})$

Step 10.1.6.2

Factor $3$ out of $6$ .

$\frac{3 (2)}{e^{\frac{1}{3}}} \cdot \frac{- 1}{3} + 5 (\frac{1}{e^{\frac{1}{3}}})$

Step 10.1.6.3

Cancel the common factor.

$\frac{3 \cdot 2}{e^{\frac{1}{3}}} \cdot \frac{- 1}{3} + 5 (\frac{1}{e^{\frac{1}{3}}})$

Step 10.1.6.4

Rewrite the expression.

$\frac{2}{e^{\frac{1}{3}}} \cdot - 1 + 5 (\frac{1}{e^{\frac{1}{3}}})$

Step 10.1.7

Combine $\frac{2}{e^{\frac{1}{3}}}$ and $- 1$ .

$\frac{2 \cdot - 1}{e^{\frac{1}{3}}} + 5 (\frac{1}{e^{\frac{1}{3}}})$

Step 10.1.8

Multiply $2$ by $- 1$ .

$\frac{- 2}{e^{\frac{1}{3}}} + 5 (\frac{1}{e^{\frac{1}{3}}})$

Step 10.1.9

Move the negative in front of the fraction.

$- \frac{2}{e^{\frac{1}{3}}} + 5 (\frac{1}{e^{\frac{1}{3}}})$

Step 10.1.10

Combine $5$ and $\frac{1}{e^{\frac{1}{3}}}$ .

$- \frac{2}{e^{\frac{1}{3}}} + \frac{5}{e^{\frac{1}{3}}}$

Step 10.2

Combine fractions.

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Step 10.2.1

Combine the numerators over the common denominator.

$\frac{- 2 + 5}{e^{\frac{1}{3}}}$

Step 10.2.2

Add $- 2$ and $5$ .

$\frac{3}{e^{\frac{1}{3}}}$

Step 11

$x = \frac{1}{e^{\frac{1}{3}}}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{1}{e^{\frac{1}{3}}}$ is a local minimum

Step 12

Find the y-value when $x = \frac{1}{e^{\frac{1}{3}}}$ .

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Step 12.1

Replace the variable $x$ with $\frac{1}{e^{\frac{1}{3}}}$ in the expression.

$f (\frac{1}{e^{\frac{1}{3}}}) = {(\frac{1}{e^{\frac{1}{3}}})}^{3} \ln (\frac{1}{e^{\frac{1}{3}}})$

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify the expression.

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Step 12.2.1.1

Apply the product rule to $\frac{1}{e^{\frac{1}{3}}}$ .

$f (\frac{1}{e^{\frac{1}{3}}}) = \frac{1^{3}}{{(e^{\frac{1}{3}})}^{3}} \cdot \ln (\frac{1}{e^{\frac{1}{3}}})$

Step 12.2.1.2

One to any power is one.

$f (\frac{1}{e^{\frac{1}{3}}}) = \frac{1}{{(e^{\frac{1}{3}})}^{3}} \cdot \ln (\frac{1}{e^{\frac{1}{3}}})$

Step 12.2.2

Simplify the denominator.

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Step 12.2.2.1

Multiply the exponents in ${(e^{\frac{1}{3}})}^{3}$ .

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Step 12.2.2.1.1

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

$f (\frac{1}{e^{\frac{1}{3}}}) = \frac{1}{e^{\frac{1}{3} \cdot 3}} \cdot \ln (\frac{1}{e^{\frac{1}{3}}})$

Step 12.2.2.1.2

Cancel the common factor of $3$ .

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Step 12.2.2.1.2.1

Cancel the common factor.

$f (\frac{1}{e^{\frac{1}{3}}}) = \frac{1}{e^{\frac{1}{3} \cdot 3}} \cdot \ln (\frac{1}{e^{\frac{1}{3}}})$

Step 12.2.2.1.2.2

Rewrite the expression.

$f (\frac{1}{e^{\frac{1}{3}}}) = \frac{1}{e} \cdot \ln (\frac{1}{e^{\frac{1}{3}}})$

Step 12.2.2.2

Simplify.

$f (\frac{1}{e^{\frac{1}{3}}}) = \frac{1}{e} \cdot \ln (\frac{1}{e^{\frac{1}{3}}})$

Step 12.2.3

Move $e^{\frac{1}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$f (\frac{1}{e^{\frac{1}{3}}}) = \frac{1}{e} \cdot \ln (e^{- \frac{1}{3}})$

Step 12.2.4

Expand $\ln (e^{- \frac{1}{3}})$ by moving $- \frac{1}{3}$ outside the logarithm.

$f (\frac{1}{e^{\frac{1}{3}}}) = \frac{1}{e} \cdot (- \frac{1}{3} \cdot \ln (e))$

Step 12.2.5

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

$f (\frac{1}{e^{\frac{1}{3}}}) = \frac{1}{e} \cdot (- \frac{1}{3} \cdot 1)$

Step 12.2.6

Multiply $- 1$ by $1$ .

$f (\frac{1}{e^{\frac{1}{3}}}) = \frac{1}{e} \cdot (- \frac{1}{3})$

Step 12.2.7

Multiply $\frac{1}{e}$ by $\frac{1}{3}$ .

$f (\frac{1}{e^{\frac{1}{3}}}) = - \frac{1}{e \cdot 3}$

Step 12.2.8

Move $3$ to the left of $e$ .

$f (\frac{1}{e^{\frac{1}{3}}}) = - \frac{1}{3 e}$

Step 12.2.9

The final answer is $- \frac{1}{3 e}$ .

$y = - \frac{1}{3 e}$

Step 13

These are the local extrema for $f (x) = x^{3} \ln (x)$ .

$(\frac{1}{e^{\frac{1}{3}}}, - \frac{1}{3 e})$ is a local minima

Step 14