Calculus Examples

$y = \frac{e^{x}}{x}$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = e^{x}$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [x]}{x^{2}}$

Step 1.2

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

$\frac{x e^{x} - e^{x} \frac{d}{d x} [x]}{x^{2}}$

Step 1.3

Differentiate using the Power Rule.

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Step 1.3.1

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

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

Step 1.3.2

Multiply $- 1$ by $1$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Reorder terms.

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

Step 1.4.2

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

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Step 1.4.2.1

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

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

Step 1.4.2.2

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

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

Step 1.4.2.3

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

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

Step 2

Find the second derivative of the function.

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Step 2.1

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

Step 2.2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 2.2.1

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

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

Step 2.2.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{x^{2} \frac{d}{d x} (e^{x} (x - 1)) - e^{x} (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.3

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 $f (x) = e^{x}$ and $g (x) = x - 1$ .

$f'' (x) = \frac{x^{2} (e^{x} \frac{d}{d x} (x - 1) + (x - 1) \frac{d}{d x} (e^{x})) - e^{x} (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.4

Differentiate.

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Step 2.4.1

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$f'' (x) = \frac{x^{2} (e^{x} (\frac{d}{d x} (x) + \frac{d}{d x} (- 1)) + (x - 1) \frac{d}{d x} (e^{x})) - e^{x} (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.4.2

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

$f'' (x) = \frac{x^{2} (e^{x} (1 + \frac{d}{d x} (- 1)) + (x - 1) \frac{d}{d x} (e^{x})) - e^{x} (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.4.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$f'' (x) = \frac{x^{2} (e^{x} (1 + 0) + (x - 1) \frac{d}{d x} (e^{x})) - e^{x} (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.4.4

Simplify the expression.

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Step 2.4.4.1

Add $1$ and $0$ .

$f'' (x) = \frac{x^{2} (e^{x} \cdot 1 + (x - 1) \frac{d}{d x} (e^{x})) - e^{x} (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.4.4.2

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

$f'' (x) = \frac{x^{2} (e^{x} + (x - 1) \frac{d}{d x} (e^{x})) - e^{x} (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.5

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

$f'' (x) = \frac{x^{2} (e^{x} + (x - 1) e^{x}) - e^{x} (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.6

Differentiate using the Power Rule.

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Step 2.6.1

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

$f'' (x) = \frac{x^{2} (e^{x} + (x - 1) e^{x}) - e^{x} (x - 1) (2 x)}{x^{4}}$

Step 2.6.2

Simplify with factoring out.

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Step 2.6.2.1

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{x^{2} (e^{x} + (x - 1) e^{x}) - 2 e^{x} (x - 1) x}{x^{4}}$

Step 2.6.2.2

Factor $x$ out of $x^{2} (e^{x} + (x - 1) e^{x}) - 2 e^{x} (x - 1) x$ .

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Step 2.6.2.2.1

Factor $x$ out of $x^{2} (e^{x} + (x - 1) e^{x})$ .

$f'' (x) = \frac{x (x (e^{x} + (x - 1) e^{x})) - 2 e^{x} (x - 1) x}{x^{4}}$

Step 2.6.2.2.2

Factor $x$ out of $- 2 e^{x} (x - 1) x$ .

$f'' (x) = \frac{x (x (e^{x} + (x - 1) e^{x})) + x (- 2 e^{x} (x - 1))}{x^{4}}$

Step 2.6.2.2.3

Factor $x$ out of $x (x (e^{x} + (x - 1) e^{x})) + x (- 2 e^{x} (x - 1))$ .

$f'' (x) = \frac{x (x (e^{x} + (x - 1) e^{x}) - 2 e^{x} (x - 1))}{x^{4}}$

Step 2.7

Cancel the common factors.

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Step 2.7.1

Factor $x$ out of $x^{4}$ .

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

Step 2.7.2

Cancel the common factor.

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

Step 2.7.3

Rewrite the expression.

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

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

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

Step 2.8.2

Apply the distributive property.

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

Step 2.8.3

Apply the distributive property.

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

Step 2.8.4

Simplify the numerator.

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Step 2.8.4.1

Combine the opposite terms in $x e^{x} + x (x e^{x}) + x (- 1 e^{x}) - 2 e^{x} x - 2 e^{x} \cdot - 1$ .

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Step 2.8.4.1.1

Reorder the factors in the terms $x e^{x}$ and $x (- 1 e^{x})$ .

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

Step 2.8.4.1.2

Subtract $e^{x} x$ from $e^{x} x$ .

$f'' (x) = \frac{x (x e^{x}) + 0 - 2 e^{x} x - 2 e^{x} \cdot - 1}{x^{3}}$

Step 2.8.4.1.3

Add $x (x e^{x})$ and $0$ .

$f'' (x) = \frac{x (x e^{x}) - 2 e^{x} x - 2 e^{x} \cdot - 1}{x^{3}}$

Step 2.8.4.2

Simplify each term.

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Step 2.8.4.2.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.8.4.2.1.1

Move $x$ .

$f'' (x) = \frac{x \cdot x e^{x} - 2 e^{x} x - 2 e^{x} \cdot - 1}{x^{3}}$

Step 2.8.4.2.1.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{x^{2} e^{x} - 2 e^{x} x - 2 e^{x} \cdot - 1}{x^{3}}$

Step 2.8.4.2.2

Multiply $- 1$ by $- 2$ .

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

Step 2.8.4.3

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

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

Step 2.8.5

Reorder terms.

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

Step 2.8.6

Reorder factors in $\frac{e^{x} x^{2} - 2 e^{x} x + 2 e^{x}}{x^{3}}$ .

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

Step 3

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

$\frac{e^{x} (x - 1)}{x^{2}} = 0$

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

$\frac{x \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [x]}{x^{2}}$

Step 4.1.2

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

$\frac{x e^{x} - e^{x} \frac{d}{d x} [x]}{x^{2}}$

Step 4.1.3

Differentiate using the Power Rule.

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Step 4.1.3.1

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

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

Step 4.1.3.2

Multiply $- 1$ by $1$ .

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

Step 4.1.4

Simplify.

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Step 4.1.4.1

Reorder terms.

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

Step 4.1.4.2

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

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Step 4.1.4.2.1

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

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

Step 4.1.4.2.2

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

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

Step 4.1.4.2.3

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

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

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{e^{x} (x - 1)}{x^{2}}$ .

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

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{e^{x} (x - 1)}{x^{2}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{e^{x} (x - 1)}{x^{2}} = 0$

Step 5.2

Set the numerator equal to zero.

$e^{x} (x - 1) = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

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

$e^{x} = 0$

$x - 1 = 0$

Step 5.3.2

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

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Step 5.3.2.1

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

$e^{x} = 0$

Step 5.3.2.2

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

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Step 5.3.2.2.1

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

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

Step 5.3.2.2.2

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

Undefined

Step 5.3.2.2.3

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

No solution

Step 5.3.3

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 5.3.3.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 5.3.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 5.3.4

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

$x = 1$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{e^{x} (x - 1)}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 6.2

Solve for $x$ .

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Step 6.2.1

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

$x = \pm \sqrt{0}$

Step 6.2.2

Simplify $\pm \sqrt{0}$ .

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Step 6.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 6.2.2.2

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

$x = \pm 0$

Step 6.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 7

Critical points to evaluate.

$x = 1$

Step 8

Evaluate the second derivative at $x = 1$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

One to any power is one.

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

Step 9.1.2

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

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

Step 9.1.3

Simplify.

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

Step 9.1.4

Multiply $- 2$ by $1$ .

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

Step 9.1.5

Simplify.

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

Step 9.1.6

Simplify.

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

Step 9.1.7

Subtract $2 e$ from $e$ .

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

Step 9.1.8

Add $- e$ and $2 e$ .

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

Step 9.2

Simplify the expression.

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Step 9.2.1

One to any power is one.

$\frac{e}{1}$

Step 9.2.2

Divide $e$ by $1$ .

$e$

Step 10

$x = 1$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 1$ is a local minimum

Step 11

Find the y-value when $x = 1$ .

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Step 11.1

Replace the variable $x$ with $1$ in the expression.

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

Step 11.2

Simplify the result.

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Step 11.2.1

Divide $e^{1}$ by $1$ .

$f (1) = e$

Step 11.2.2

The final answer is $e$ .

$y = e$

Step 12

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

$(1, e)$ is a local minima

Step 13