Calculus Examples

$f (x) = x e^{k x}$

Step 1

Find the first derivative of the function.

Tap for more steps...

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 $f (x) = x$ and $g (x) = e^{k x}$ .

$x \frac{d}{d x} [e^{k x}] + e^{k x} \frac{d}{d x} [x]$

Step 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 $g (x) = k x$ .

Tap for more steps...

Step 1.2.1

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

$x (\frac{d}{d u} [e^{u}] \frac{d}{d x} [k x]) + e^{k x} \frac{d}{d x} [x]$

Step 1.2.2

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

$x (e^{u} \frac{d}{d x} [k x]) + e^{k x} \frac{d}{d x} [x]$

Step 1.2.3

Replace all occurrences of $u$ with $k x$ .

$x (e^{k x} \frac{d}{d x} [k x]) + e^{k x} \frac{d}{d x} [x]$

Step 1.3

Differentiate.

Tap for more steps...

Step 1.3.1

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

$x (e^{k x} (k \frac{d}{d x} [x])) + e^{k x} \frac{d}{d x} [x]$

Step 1.3.2

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

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

Step 1.3.3

Multiply $k$ by $1$ .

$x (e^{k x} k) + e^{k x} \frac{d}{d x} [x]$

Step 1.3.4

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

$x (e^{k x} k) + e^{k x} \cdot 1$

Step 1.3.5

Multiply $e^{k x}$ by $1$ .

$x (e^{k x} k) + e^{k x}$

Step 1.4

Simplify.

Tap for more steps...

Step 1.4.1

Reorder terms.

$e^{k x} k x + e^{k x}$

Step 1.4.2

Reorder factors in $e^{k x} k x + e^{k x}$ .

$k x e^{k x} + e^{k x}$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

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

$f'' (x) = \frac{d}{d x} (k x e^{k x}) + \frac{d}{d x} (e^{k x})$

Step 2.2

Evaluate $\frac{d}{d x} [k x e^{k x}]$ .

Tap for more steps...

Step 2.2.1

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

$f'' (x) = k \frac{d}{d x} (x e^{k x}) + \frac{d}{d x} (e^{k x})$

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 $g (x) = e^{k x}$ .

$f'' (x) = k (x \frac{d}{d x} (e^{k x}) + e^{k x} \frac{d}{d x} (x)) + \frac{d}{d x} (e^{k x})$

Step 2.2.3

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 $g (x) = k x$ .

Tap for more steps...

Step 2.2.3.1

To apply the Chain Rule, set $u_{1}$ as $k x$ .

$f'' (x) = k (x (\frac{d}{d u (1)} (e^{u_{1}}) \frac{d}{d x} (k x)) + e^{k x} \frac{d}{d x} (x)) + \frac{d}{d x} (e^{k x})$

Step 2.2.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$f'' (x) = k (x (e^{u_{1}} \frac{d}{d x} (k x)) + e^{k x} \frac{d}{d x} (x)) + \frac{d}{d x} (e^{k x})$

Step 2.2.3.3

Replace all occurrences of $u_{1}$ with $k x$ .

$f'' (x) = k (x (e^{k x} \frac{d}{d x} (k x)) + e^{k x} \frac{d}{d x} (x)) + \frac{d}{d x} (e^{k x})$

Step 2.2.4

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

$f'' (x) = k (x (e^{k x} (k \frac{d}{d x} (x))) + e^{k x} \frac{d}{d x} (x)) + \frac{d}{d x} (e^{k x})$

Step 2.2.5

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

$f'' (x) = k (x (e^{k x} (k \cdot 1)) + e^{k x} \frac{d}{d x} (x)) + \frac{d}{d x} (e^{k x})$

Step 2.2.6

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

$f'' (x) = k (x (e^{k x} (k \cdot 1)) + e^{k x} \cdot 1) + \frac{d}{d x} (e^{k x})$

Step 2.2.7

Multiply $k$ by $1$ .

$f'' (x) = k (x (e^{k x} k) + e^{k x} \cdot 1) + \frac{d}{d x} (e^{k x})$

Step 2.2.8

Multiply $e^{k x}$ by $1$ .

$f'' (x) = k (x (e^{k x} k) + e^{k x}) + \frac{d}{d x} (e^{k x})$

Step 2.3

Evaluate $\frac{d}{d x} [e^{k x}]$ .

Tap for more steps...

Step 2.3.1

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 $g (x) = k x$ .

Tap for more steps...

Step 2.3.1.1

To apply the Chain Rule, set $u_{2}$ as $k x$ .

$f'' (x) = k (x (e^{k x} k) + e^{k x}) + \frac{d}{d u (2)} (e^{u_{2}}) \frac{d}{d x} (k x)$

Step 2.3.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$f'' (x) = k (x (e^{k x} k) + e^{k x}) + e^{u_{2}} \frac{d}{d x} (k x)$

Step 2.3.1.3

Replace all occurrences of $u_{2}$ with $k x$ .

$f'' (x) = k (x (e^{k x} k) + e^{k x}) + e^{k x} \frac{d}{d x} (k x)$

Step 2.3.2

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

$f'' (x) = k (x (e^{k x} k) + e^{k x}) + e^{k x} (k \frac{d}{d x} (x))$

Step 2.3.3

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

$f'' (x) = k (x (e^{k x} k) + e^{k x}) + e^{k x} (k \cdot 1)$

Step 2.3.4

Multiply $k$ by $1$ .

$f'' (x) = k (x (e^{k x} k) + e^{k x}) + e^{k x} k$

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Apply the distributive property.

$f'' (x) = k (x (e^{k x} k)) + k e^{k x} + e^{k x} k$

Step 2.4.2

Combine terms.

Tap for more steps...

Step 2.4.2.1

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

$f'' (x) = k \cdot k (x (e^{k x})) + k e^{k x} + e^{k x} k$

Step 2.4.2.2

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

$f'' (x) = k \cdot k (x (e^{k x})) + k e^{k x} + e^{k x} k$

Step 2.4.2.3

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

$f'' (x) = k^{1 + 1} (x (e^{k x})) + k e^{k x} + e^{k x} k$

Step 2.4.2.4

Add $1$ and $1$ .

$f'' (x) = k^{2} (x (e^{k x})) + k e^{k x} + e^{k x} k$

Step 2.4.2.5

Add $k e^{k x}$ and $e^{k x} k$ .

Tap for more steps...

Step 2.4.2.5.1

Reorder $e^{k x}$ and $k$ .

$f'' (x) = k^{2} (x (e^{k x})) + k e^{k x} + k e^{k x}$

Step 2.4.2.5.2

Add $k e^{k x}$ and $k e^{k x}$ .

$f'' (x) = k^{2} (x (e^{k x})) + 2 k e^{k x}$

Step 2.4.3

Reorder terms.

$f'' (x) = e^{k x} k^{2} x + 2 e^{k x} k$

Step 2.4.4

Reorder factors in $e^{k x} k^{2} x + 2 e^{k x} k$ .

$f'' (x) = k^{2} x e^{k x} + 2 k e^{k x}$

Step 3

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

$k x e^{k x} + e^{k x} = 0$

Step 4

Find the first derivative.

Tap for more steps...

Step 4.1

Find the first derivative.

Tap for more steps...

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 $f (x) = x$ and $g (x) = e^{k x}$ .

$x \frac{d}{d x} [e^{k x}] + e^{k x} \frac{d}{d x} [x]$

Step 4.1.2

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 $g (x) = k x$ .

Tap for more steps...

Step 4.1.2.1

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

$x (\frac{d}{d u} [e^{u}] \frac{d}{d x} [k x]) + e^{k x} \frac{d}{d x} [x]$

Step 4.1.2.2

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

$x (e^{u} \frac{d}{d x} [k x]) + e^{k x} \frac{d}{d x} [x]$

Step 4.1.2.3

Replace all occurrences of $u$ with $k x$ .

$x (e^{k x} \frac{d}{d x} [k x]) + e^{k x} \frac{d}{d x} [x]$

Step 4.1.3

Differentiate.

Tap for more steps...

Step 4.1.3.1

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

$x (e^{k x} (k \frac{d}{d x} [x])) + e^{k x} \frac{d}{d x} [x]$

Step 4.1.3.2

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

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

Step 4.1.3.3

Multiply $k$ by $1$ .

$x (e^{k x} k) + e^{k x} \frac{d}{d x} [x]$

Step 4.1.3.4

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

$x (e^{k x} k) + e^{k x} \cdot 1$

Step 4.1.3.5

Multiply $e^{k x}$ by $1$ .

$x (e^{k x} k) + e^{k x}$

Step 4.1.4

Simplify.

Tap for more steps...

Step 4.1.4.1

Reorder terms.

$e^{k x} k x + e^{k x}$

Step 4.1.4.2

Reorder factors in $e^{k x} k x + e^{k x}$ .

$f' (x) = k x e^{k x} + e^{k x}$

Step 4.2

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

$k x e^{k x} + e^{k x}$

Step 5

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

Tap for more steps...

Step 5.1

Set the first derivative equal to $0$ .

$k x e^{k x} + e^{k x} = 0$

Step 5.2

Factor $e^{k x}$ out of $k x e^{k x} + e^{k x}$ .

Tap for more steps...

Step 5.2.1

Factor $e^{k x}$ out of $k x e^{k x}$ .

$e^{k x} (k x) + e^{k x} = 0$

Step 5.2.2

Multiply by $1$ .

$e^{k x} (k x) + e^{k x} \cdot 1 = 0$

Step 5.2.3

Factor $e^{k x}$ out of $e^{k x} (k x) + e^{k x} \cdot 1$ .

$e^{k x} (k x + 1) = 0$

Step 5.3

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

$e^{k x} = 0$

$k x + 1 = 0$

Step 5.4

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

Tap for more steps...

Step 5.4.1

Set $e^{k x}$ equal to $0$ .

$e^{k x} = 0$

Step 5.4.2

Solve $e^{k x} = 0$ for $x$ .

Tap for more steps...

Step 5.4.2.1

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

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

Step 5.4.2.2

Expand the left side.

Tap for more steps...

Step 5.4.2.2.1

Expand $\ln (e^{k x})$ by moving $k x$ outside the logarithm.

$k x \ln (e) = \ln (0)$

Step 5.4.2.2.2

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

$k x \cdot 1 = \ln (0)$

Step 5.4.2.2.3

Multiply $k$ by $1$ .

$k x = \ln (0)$

Step 5.4.2.3

Simplify the right side.

Tap for more steps...

Step 5.4.2.3.1

The equation cannot be solved because it is undefined.

$Undefined$

Step 5.4.2.4

Divide each term in $k x = Undefined$ by $k$ and simplify.

Tap for more steps...

Step 5.4.2.4.1

Divide each term in $k x = Undefined$ by $k$ .

$\frac{k x}{k} = \frac{Undefined}{k}$

Step 5.4.2.4.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.4.2.1

Cancel the common factor of $k$ .

Tap for more steps...

Step 5.4.2.4.2.1.1

Cancel the common factor.

$\frac{k x}{k} = \frac{Undefined}{k}$

Step 5.4.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{Undefined}{k}$

Step 5.5

Set $k x + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.5.1

Set $k x + 1$ equal to $0$ .

$k x + 1 = 0$

Step 5.5.2

Solve $k x + 1 = 0$ for $x$ .

Tap for more steps...

Step 5.5.2.1

Subtract $1$ from both sides of the equation.

$k x = - 1$

Step 5.5.2.2

Divide each term in $k x = - 1$ by $k$ and simplify.

Tap for more steps...

Step 5.5.2.2.1

Divide each term in $k x = - 1$ by $k$ .

$\frac{k x}{k} = \frac{- 1}{k}$

Step 5.5.2.2.2

Simplify the left side.

Tap for more steps...

Step 5.5.2.2.2.1

Cancel the common factor of $k$ .

Tap for more steps...

Step 5.5.2.2.2.1.1

Cancel the common factor.

$\frac{k x}{k} = \frac{- 1}{k}$

Step 5.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{k}$

Step 5.5.2.2.3

Simplify the right side.

Tap for more steps...

Step 5.5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{k}$

Step 5.6

The final solution is all the values that make $e^{k x} (k x + 1) = 0$ true.

$x = - \frac{1}{k}$

Step 6

Find the values where the derivative is undefined.

Tap for more steps...

Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = - \frac{1}{k}$

Step 8

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

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

Step 9

Evaluate the second derivative.

Tap for more steps...

Step 9.1

Simplify each term.

Tap for more steps...

Step 9.1.1

Rewrite using the commutative property of multiplication.

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

Step 9.1.2

Cancel the common factor of $k$ .

Tap for more steps...

Step 9.1.2.1

Factor $k$ out of $- k^{2}$ .

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

Step 9.1.2.2

Cancel the common factor.

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

Step 9.1.2.3

Rewrite the expression.

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

Step 9.1.3

Rewrite using the commutative property of multiplication.

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

Step 9.1.4

Cancel the common factor of $k$ .

Tap for more steps...

Step 9.1.4.1

Factor $k$ out of $- k$ .

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

Step 9.1.4.2

Cancel the common factor.

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

Step 9.1.4.3

Rewrite the expression.

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

Step 9.1.5

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

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

Step 9.1.6

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

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

Step 9.1.7

Rewrite using the commutative property of multiplication.

$- \frac{k}{e} + 2 k e^{- k \frac{1}{k}}$

Step 9.1.8

Cancel the common factor of $k$ .

Tap for more steps...

Step 9.1.8.1

Factor $k$ out of $- k$ .

$- \frac{k}{e} + 2 k e^{k \cdot - 1 \frac{1}{k}}$

Step 9.1.8.2

Cancel the common factor.

$- \frac{k}{e} + 2 k e^{k \cdot - 1 \frac{1}{k}}$

Step 9.1.8.3

Rewrite the expression.

$- \frac{k}{e} + 2 k e^{- 1}$

Step 9.1.9

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

$- \frac{k}{e} + 2 k \frac{1}{e}$

Step 9.1.10

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

Tap for more steps...

Step 9.1.10.1

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

$- \frac{k}{e} + k \frac{2}{e}$

Step 9.1.10.2

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

$- \frac{k}{e} + \frac{k \cdot 2}{e}$

Step 9.1.11

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

$- \frac{k}{e} + \frac{2 k}{e}$

Step 9.2

Simplify terms.

Tap for more steps...

Step 9.2.1

Combine the numerators over the common denominator.

$\frac{- k + 2 k}{e}$

Step 9.2.2

Add $- k$ and $2 k$ .

$\frac{k}{e}$

Step 10

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

Tap for more steps...

Step 10.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, \infty)$

Step 10.2

Substitute any number, such as $0$ , from the interval $(- \infty, \infty)$ in the first derivative $k x e^{k x} + e^{k x}$ to check if the result is negative or positive.

Tap for more steps...

Step 10.2.1

Replace the variable $x$ with $0$ in the expression.

$f' (0) = k (0) \cdot e^{k (0)} + e^{k (0)}$

Step 10.2.2

Simplify the result.

Tap for more steps...

Step 10.2.2.1

Simplify each term.

Tap for more steps...

Step 10.2.2.1.1

Multiply $k$ by $0$ .

$f' (0) = 0 \cdot e^{k (0)} + e^{k (0)}$

Step 10.2.2.1.2

Multiply $k$ by $0$ .

$f' (0) = 0 \cdot e^{0} + e^{k (0)}$

Step 10.2.2.1.3

Anything raised to $0$ is $1$ .

$f' (0) = 0 \cdot 1 + e^{k (0)}$

Step 10.2.2.1.4

Multiply $0$ by $1$ .

$f' (0) = 0 + e^{k (0)}$

Step 10.2.2.1.5

Multiply $k$ by $0$ .

$f' (0) = 0 + e^{0}$

Step 10.2.2.1.6

Anything raised to $0$ is $1$ .

$f' (0) = 0 + 1$

Step 10.2.2.2

Add $0$ and $1$ .

$f' (0) = 1$

Step 10.2.2.3

The final answer is $1$ .

$1$

Step 10.3

No local maxima or minima found for $f (x) = x e^{k x}$ .

No local maxima or minima

Step 11