Calculus Examples

$f (x) = (x - 4) e^{- 3 x}$

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 $f (x) = x - 4$ and $g (x) = e^{- 3 x}$ .

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

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - 3 x$ .

Tap for more steps...

Step 1.1.2.1

To apply the Chain Rule, set $u$ as $- 3 x$ .

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

Step 1.1.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

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

Step 1.1.2.3

Replace all occurrences of $u$ with $- 3 x$ .

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

Step 1.1.3

Differentiate.

Tap for more steps...

Step 1.1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Simplify the expression.

Tap for more steps...

Step 1.1.3.3.1

Multiply $- 3$ by $1$ .

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

Step 1.1.3.3.2

Move $- 3$ to the left of $(x - 4) e^{- 3 x}$ .

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

Step 1.1.3.4

By the Sum Rule, the derivative of $x - 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

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

Step 1.1.3.5

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

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

Step 1.1.3.6

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$- 3 (x - 4) e^{- 3 x} + e^{- 3 x} (1 + 0)$

Step 1.1.3.7

Simplify the expression.

Tap for more steps...

Step 1.1.3.7.1

Add $1$ and $0$ .

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

Step 1.1.3.7.2

Multiply $e^{- 3 x}$ by $1$ .

$- 3 (x - 4) e^{- 3 x} + e^{- 3 x}$

Step 1.1.4

Simplify.

Tap for more steps...

Step 1.1.4.1

Apply the distributive property.

$(- 3 x - 3 \cdot - 4) e^{- 3 x} + e^{- 3 x}$

Step 1.1.4.2

Apply the distributive property.

$- 3 x e^{- 3 x} - 3 \cdot - 4 e^{- 3 x} + e^{- 3 x}$

Step 1.1.4.3

Combine terms.

Tap for more steps...

Step 1.1.4.3.1

Multiply $- 3$ by $- 4$ .

$- 3 x e^{- 3 x} + 12 e^{- 3 x} + e^{- 3 x}$

Step 1.1.4.3.2

Add $12 e^{- 3 x}$ and $e^{- 3 x}$ .

$- 3 x e^{- 3 x} + 13 e^{- 3 x}$

Step 1.1.4.4

Reorder terms.

$- 3 e^{- 3 x} x + 13 e^{- 3 x}$

Step 1.1.4.5

Reorder factors in $- 3 e^{- 3 x} x + 13 e^{- 3 x}$ .

$f' (x) = - 3 x e^{- 3 x} + 13 e^{- 3 x}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- 3 x e^{- 3 x} + 13 e^{- 3 x}$ .

$- 3 x e^{- 3 x} + 13 e^{- 3 x}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- 3 x e^{- 3 x} + 13 e^{- 3 x} = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$- 3 x e^{- 3 x} + 13 e^{- 3 x} = 0$

Step 2.2

Factor $e^{- 3 x}$ out of $- 3 x e^{- 3 x} + 13 e^{- 3 x}$ .

Tap for more steps...

Step 2.2.1

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

$e^{- 3 x} (- 3 x) + 13 e^{- 3 x} = 0$

Step 2.2.2

Factor $e^{- 3 x}$ out of $13 e^{- 3 x}$ .

$e^{- 3 x} (- 3 x) + e^{- 3 x} \cdot 13 = 0$

Step 2.2.3

Factor $e^{- 3 x}$ out of $e^{- 3 x} (- 3 x) + e^{- 3 x} \cdot 13$ .

$e^{- 3 x} (- 3 x + 13) = 0$

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$e^{- 3 x} = 0$

$- 3 x + 13 = 0$

Step 2.4

Set $e^{- 3 x}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.4.1

Set $e^{- 3 x}$ equal to $0$ .

$e^{- 3 x} = 0$

Step 2.4.2

Solve $e^{- 3 x} = 0$ for $x$ .

Tap for more steps...

Step 2.4.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{- 3 x}) = \ln (0)$

Step 2.4.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 2.4.2.3

There is no solution for $e^{- 3 x} = 0$

No solution

Step 2.5

Set $- 3 x + 13$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.5.1

Set $- 3 x + 13$ equal to $0$ .

$- 3 x + 13 = 0$

Step 2.5.2

Solve $- 3 x + 13 = 0$ for $x$ .

Tap for more steps...

Step 2.5.2.1

Subtract $13$ from both sides of the equation.

$- 3 x = - 13$

Step 2.5.2.2

Divide each term in $- 3 x = - 13$ by $- 3$ and simplify.

Tap for more steps...

Step 2.5.2.2.1

Divide each term in $- 3 x = - 13$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 13}{- 3}$

Step 2.5.2.2.2

Simplify the left side.

Tap for more steps...

Step 2.5.2.2.2.1

Cancel the common factor of $- 3$ .

Tap for more steps...

Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 13}{- 3}$

Step 2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 13}{- 3}$

Step 2.5.2.2.3

Simplify the right side.

Tap for more steps...

Step 2.5.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{13}{3}$

Step 2.6

The final solution is all the values that make $e^{- 3 x} (- 3 x + 13) = 0$ true.

$x = \frac{13}{3}$

Step 3

The values which make the derivative equal to $0$ are $\frac{13}{3}$ .

$\frac{13}{3}$

Step 4

After finding the point that makes the derivative $f' (x) = - 3 x e^{- 3 x} + 13 e^{- 3 x}$ equal to $0$ or undefined, the interval to check where $f (x) = (x - 4) e^{- 3 x}$ is increasing and where it is decreasing is $(- \infty, \frac{13}{3}) \cup (\frac{13}{3}, \infty)$ .

$(- \infty, \frac{13}{3}) \cup (\frac{13}{3}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, \frac{13}{3})$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 5.1

Replace the variable $x$ with $\frac{10}{3}$ in the expression.

$f' (\frac{10}{3}) = - 3 (\frac{10}{3}) \cdot e^{- 3 (\frac{10}{3})} + 13 e^{- 3 (\frac{10}{3})}$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.1.1.1

Factor $3$ out of $- 3$ .

$f' (\frac{10}{3}) = 3 (- 1) (\frac{10}{3}) \cdot e^{- 3 (\frac{10}{3})} + 13 e^{- 3 (\frac{10}{3})}$

Step 5.2.1.1.2

Cancel the common factor.

$f' (\frac{10}{3}) = 3 \cdot (- 1 (\frac{10}{3})) \cdot e^{- 3 (\frac{10}{3})} + 13 e^{- 3 (\frac{10}{3})}$

Step 5.2.1.1.3

Rewrite the expression.

$f' (\frac{10}{3}) = - 1 \cdot 10 \cdot e^{- 3 (\frac{10}{3})} + 13 e^{- 3 (\frac{10}{3})}$

Step 5.2.1.2

Multiply $- 1$ by $10$ .

$f' (\frac{10}{3}) = - 10 \cdot e^{- 3 (\frac{10}{3})} + 13 e^{- 3 (\frac{10}{3})}$

Step 5.2.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.1.3.1

Factor $3$ out of $- 3$ .

$f' (\frac{10}{3}) = - 10 \cdot e^{3 (- 1) (\frac{10}{3})} + 13 e^{- 3 (\frac{10}{3})}$

Step 5.2.1.3.2

Cancel the common factor.

$f' (\frac{10}{3}) = - 10 \cdot e^{3 \cdot (- 1 (\frac{10}{3}))} + 13 e^{- 3 (\frac{10}{3})}$

Step 5.2.1.3.3

Rewrite the expression.

$f' (\frac{10}{3}) = - 10 \cdot e^{- 1 \cdot 10} + 13 e^{- 3 (\frac{10}{3})}$

Step 5.2.1.4

Multiply $- 1$ by $10$ .

$f' (\frac{10}{3}) = - 10 \cdot e^{- 10} + 13 e^{- 3 (\frac{10}{3})}$

Step 5.2.1.5

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

$f' (\frac{10}{3}) = - 10 \cdot \frac{1}{e^{10}} + 13 e^{- 3 (\frac{10}{3})}$

Step 5.2.1.6

Combine $- 10$ and $\frac{1}{e^{10}}$ .

$f' (\frac{10}{3}) = \frac{- 10}{e^{10}} + 13 e^{- 3 (\frac{10}{3})}$

Step 5.2.1.7

Move the negative in front of the fraction.

$f' (\frac{10}{3}) = - \frac{10}{e^{10}} + 13 e^{- 3 (\frac{10}{3})}$

Step 5.2.1.8

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.1.8.1

Factor $3$ out of $- 3$ .

$f' (\frac{10}{3}) = - \frac{10}{e^{10}} + 13 e^{3 (- 1) (\frac{10}{3})}$

Step 5.2.1.8.2

Cancel the common factor.

$f' (\frac{10}{3}) = - \frac{10}{e^{10}} + 13 e^{3 \cdot (- 1 (\frac{10}{3}))}$

Step 5.2.1.8.3

Rewrite the expression.

$f' (\frac{10}{3}) = - \frac{10}{e^{10}} + 13 e^{- 1 \cdot 10}$

Step 5.2.1.9

Multiply $- 1$ by $10$ .

$f' (\frac{10}{3}) = - \frac{10}{e^{10}} + 13 e^{- 10}$

Step 5.2.1.10

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

$f' (\frac{10}{3}) = - \frac{10}{e^{10}} + 13 (\frac{1}{e^{10}})$

Step 5.2.1.11

Combine $13$ and $\frac{1}{e^{10}}$ .

$f' (\frac{10}{3}) = - \frac{10}{e^{10}} + \frac{13}{e^{10}}$

Step 5.2.2

Combine fractions.

Tap for more steps...

Step 5.2.2.1

Combine the numerators over the common denominator.

$f' (\frac{10}{3}) = \frac{- 10 + 13}{e^{10}}$

Step 5.2.2.2

Add $- 10$ and $13$ .

$f' (\frac{10}{3}) = \frac{3}{e^{10}}$

Step 5.2.3

The final answer is $\frac{3}{e^{10}}$ .

$\frac{3}{e^{10}}$

Step 5.3

At $x = \frac{10}{3}$ the derivative is $\frac{3}{e^{10}}$ . Since this is positive, the function is increasing on $(- \infty, \frac{13}{3})$ .

Increasing on $(- \infty, \frac{13}{3})$ since $f' (x) > 0$

Step 6

Substitute a value from the interval $(\frac{13}{3}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $\frac{16}{3}$ in the expression.

$f' (\frac{16}{3}) = - 3 (\frac{16}{3}) \cdot e^{- 3 (\frac{16}{3})} + 13 e^{- 3 (\frac{16}{3})}$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.2.1.1.1

Factor $3$ out of $- 3$ .

$f' (\frac{16}{3}) = 3 (- 1) (\frac{16}{3}) \cdot e^{- 3 (\frac{16}{3})} + 13 e^{- 3 (\frac{16}{3})}$

Step 6.2.1.1.2

Cancel the common factor.

$f' (\frac{16}{3}) = 3 \cdot (- 1 (\frac{16}{3})) \cdot e^{- 3 (\frac{16}{3})} + 13 e^{- 3 (\frac{16}{3})}$

Step 6.2.1.1.3

Rewrite the expression.

$f' (\frac{16}{3}) = - 1 \cdot 16 \cdot e^{- 3 (\frac{16}{3})} + 13 e^{- 3 (\frac{16}{3})}$

Step 6.2.1.2

Multiply $- 1$ by $16$ .

$f' (\frac{16}{3}) = - 16 \cdot e^{- 3 (\frac{16}{3})} + 13 e^{- 3 (\frac{16}{3})}$

Step 6.2.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.2.1.3.1

Factor $3$ out of $- 3$ .

$f' (\frac{16}{3}) = - 16 \cdot e^{3 (- 1) (\frac{16}{3})} + 13 e^{- 3 (\frac{16}{3})}$

Step 6.2.1.3.2

Cancel the common factor.

$f' (\frac{16}{3}) = - 16 \cdot e^{3 \cdot (- 1 (\frac{16}{3}))} + 13 e^{- 3 (\frac{16}{3})}$

Step 6.2.1.3.3

Rewrite the expression.

$f' (\frac{16}{3}) = - 16 \cdot e^{- 1 \cdot 16} + 13 e^{- 3 (\frac{16}{3})}$

Step 6.2.1.4

Multiply $- 1$ by $16$ .

$f' (\frac{16}{3}) = - 16 \cdot e^{- 16} + 13 e^{- 3 (\frac{16}{3})}$

Step 6.2.1.5

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

$f' (\frac{16}{3}) = - 16 \cdot \frac{1}{e^{16}} + 13 e^{- 3 (\frac{16}{3})}$

Step 6.2.1.6

Combine $- 16$ and $\frac{1}{e^{16}}$ .

$f' (\frac{16}{3}) = \frac{- 16}{e^{16}} + 13 e^{- 3 (\frac{16}{3})}$

Step 6.2.1.7

Move the negative in front of the fraction.

$f' (\frac{16}{3}) = - \frac{16}{e^{16}} + 13 e^{- 3 (\frac{16}{3})}$

Step 6.2.1.8

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.2.1.8.1

Factor $3$ out of $- 3$ .

$f' (\frac{16}{3}) = - \frac{16}{e^{16}} + 13 e^{3 (- 1) (\frac{16}{3})}$

Step 6.2.1.8.2

Cancel the common factor.

$f' (\frac{16}{3}) = - \frac{16}{e^{16}} + 13 e^{3 \cdot (- 1 (\frac{16}{3}))}$

Step 6.2.1.8.3

Rewrite the expression.

$f' (\frac{16}{3}) = - \frac{16}{e^{16}} + 13 e^{- 1 \cdot 16}$

Step 6.2.1.9

Multiply $- 1$ by $16$ .

$f' (\frac{16}{3}) = - \frac{16}{e^{16}} + 13 e^{- 16}$

Step 6.2.1.10

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

$f' (\frac{16}{3}) = - \frac{16}{e^{16}} + 13 (\frac{1}{e^{16}})$

Step 6.2.1.11

Combine $13$ and $\frac{1}{e^{16}}$ .

$f' (\frac{16}{3}) = - \frac{16}{e^{16}} + \frac{13}{e^{16}}$

Step 6.2.2

Combine fractions.

Tap for more steps...

Step 6.2.2.1

Combine the numerators over the common denominator.

$f' (\frac{16}{3}) = \frac{- 16 + 13}{e^{16}}$

Step 6.2.2.2

Simplify the expression.

Tap for more steps...

Step 6.2.2.2.1

Add $- 16$ and $13$ .

$f' (\frac{16}{3}) = \frac{- 3}{e^{16}}$

Step 6.2.2.2.2

Move the negative in front of the fraction.

$f' (\frac{16}{3}) = - \frac{3}{e^{16}}$

Step 6.2.3

The final answer is $- \frac{3}{e^{16}}$ .

$- \frac{3}{e^{16}}$

Step 6.3

At $x = \frac{16}{3}$ the derivative is $- \frac{3}{e^{16}}$ . Since this is negative, the function is decreasing on $(\frac{13}{3}, \infty)$ .

Decreasing on $(\frac{13}{3}, \infty)$ since $f' (x) < 0$

Step 7

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- \infty, \frac{13}{3})$

Decreasing on: $(\frac{13}{3}, \infty)$

Step 8