Calculus Examples

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

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

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 1.1.2

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 1.1.3

Combine fractions.

Tap for more steps...

Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.4

Multiply $x$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 1.1.4.1

Multiply $x$ by $x^{3}$ .

Tap for more steps...

Step 1.1.4.1.1

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

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

Step 1.1.4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.1.4.2

Add $1$ and $3$ .

$\frac{1}{x^{4}} + \ln (x) \frac{d}{d x} [x^{- 3}]$

Step 1.1.5

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

$\frac{1}{x^{4}} + \ln (x) (- 3 x^{- 4})$

Step 1.1.6

Simplify.

Tap for more steps...

Step 1.1.6.1

Reorder terms.

$- 3 x^{- 4} \ln (x) + \frac{1}{x^{4}}$

Step 1.1.6.2

Simplify each term.

Tap for more steps...

Step 1.1.6.2.1

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

$- 3 \frac{1}{x^{4}} \ln (x) + \frac{1}{x^{4}}$

Step 1.1.6.2.2

Combine $- 3$ and $\frac{1}{x^{4}}$ .

$\frac{- 3}{x^{4}} \ln (x) + \frac{1}{x^{4}}$

Step 1.1.6.2.3

Move the negative in front of the fraction.

$- \frac{3}{x^{4}} \ln (x) + \frac{1}{x^{4}}$

Step 1.1.6.2.4

Combine $\ln (x)$ and $\frac{3}{x^{4}}$ .

$- \frac{\ln (x) \cdot 3}{x^{4}} + \frac{1}{x^{4}}$

Step 1.1.6.2.5

Move $3$ to the left of $\ln (x)$ .

$f' (x) = - \frac{3 \ln (x)}{x^{4}} + \frac{1}{x^{4}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{3 \ln (x)}{x^{4}} + \frac{1}{x^{4}}$ .

$- \frac{3 \ln (x)}{x^{4}} + \frac{1}{x^{4}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{3 \ln (x)}{x^{4}} + \frac{1}{x^{4}} = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$- \frac{3 \ln (x)}{x^{4}} + \frac{1}{x^{4}} = 0$

Step 2.2

Subtract $\frac{1}{x^{4}}$ from both sides of the equation.

$- \frac{3 \ln (x)}{x^{4}} = - \frac{1}{x^{4}}$

Step 2.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- 3 \ln (x) = - 1$

Step 2.4

Divide each term in $- 3 \ln (x) = - 1$ by $- 3$ and simplify.

Tap for more steps...

Step 2.4.1

Divide each term in $- 3 \ln (x) = - 1$ by $- 3$ .

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

Step 2.4.2

Simplify the left side.

Tap for more steps...

Step 2.4.2.1

Cancel the common factor of $- 3$ .

Tap for more steps...

Step 2.4.2.1.1

Cancel the common factor.

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

Step 2.4.2.1.2

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

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

Step 2.4.3

Simplify the right side.

Tap for more steps...

Step 2.4.3.1

Dividing two negative values results in a positive value.

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

Step 2.5

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

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

Step 2.6

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 2.7

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

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

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

Set the denominator in $\frac{3 \ln (x)}{x^{4}}$ equal to $0$ to find where the expression is undefined.

$x^{4} = 0$

Step 3.2

Solve for $x$ .

Tap for more steps...

Step 3.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{0}$

Step 3.2.2

Simplify $\pm \sqrt[4]{0}$ .

Tap for more steps...

Step 3.2.2.1

Rewrite $0$ as $0^{4}$ .

$x = \pm \sqrt[4]{0^{4}}$

Step 3.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3.3

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

$x \leq 0$

Step 3.4

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 $x^{- 3} \ln (x)$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = e^{\frac{1}{3}}$ .

Tap for more steps...

Step 4.1.1

Substitute $e^{\frac{1}{3}}$ for $x$ .

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

Step 4.1.2

Simplify.

Tap for more steps...

Step 4.1.2.1

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

Tap for more steps...

Step 4.1.2.1.1

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

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

Step 4.1.2.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.1.2.1.2.1

Factor $3$ out of $- 3$ .

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

Step 4.1.2.1.2.2

Cancel the common factor.

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

Step 4.1.2.1.2.3

Rewrite the expression.

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

Step 4.1.2.2

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

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

Step 4.1.2.3

Use logarithm rules to move $\frac{1}{3}$ out of the exponent.

$\frac{1}{e} (\frac{1}{3} \ln (e))$

Step 4.1.2.4

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

$\frac{1}{e} (\frac{1}{3} \cdot 1)$

Step 4.1.2.5

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

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

Step 4.1.2.6

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

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

Step 4.1.2.7

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

$\frac{1}{3 e}$

Step 4.2

Evaluate at $x = 0$ .

Tap for more steps...

Step 4.2.1

Substitute $0$ for $x$ .

${(0)}^{- 3} \ln (0)$

Step 4.2.2

The natural logarithm of zero is undefined.

Undefined

Step 4.3

List all of the points.

$(e^{\frac{1}{3}}, \frac{1}{3 e})$

Step 5