Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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

$3 \frac{d}{d x} [\frac{e^{x}}{x}]$

Step 1.2

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 $g (x) = x$ .

$3 \frac{x \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [x]}{x^{2}}$

Step 1.3

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

$3 \frac{x e^{x} - e^{x} \frac{d}{d x} [x]}{x^{2}}$

Step 1.4

Differentiate using the Power Rule.

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Step 1.4.1

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

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

Step 1.4.2

Combine fractions.

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Step 1.4.2.1

Multiply $- 1$ by $1$ .

$3 \frac{x e^{x} - e^{x}}{x^{2}}$

Step 1.4.2.2

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

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

Step 1.5

Simplify.

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Multiply $- 1$ by $3$ .

$\frac{3 x e^{x} - 3 e^{x}}{x^{2}}$

Step 1.5.3

Reorder terms.

$\frac{3 e^{x} x - 3 e^{x}}{x^{2}}$

Step 1.5.4

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

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Step 1.5.4.1

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

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

Step 1.5.4.2

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

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

Step 1.5.4.3

Factor $3 e^{x}$ out of $3 e^{x} (x) + 3 e^{x} (- 1)$ .

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

Step 2

Find the second derivative.

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Step 2.1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3 e^{x} (x - 1)}{x^{2}}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{e^{x} (x - 1)}{x^{2}}]$ .

$3 \frac{d}{d x} [\frac{e^{x} (x - 1)}{x^{2}}]$

Step 2.2

$3 \frac{x^{2} \frac{d}{d x} [e^{x} (x - 1)] - e^{x} (x - 1) \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Multiply $2$ by $2$ .

$3 \frac{x^{2} \frac{d}{d x} [e^{x} (x - 1)] - e^{x} (x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.4

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 $g (x) = x - 1$ .

$3 \frac{x^{2} (e^{x} \frac{d}{d x} [x - 1] + (x - 1) \frac{d}{d x} [e^{x}]) - e^{x} (x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.5

Differentiate.

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Step 2.5.1

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

$3 \frac{x^{2} (e^{x} (\frac{d}{d x} [x] + \frac{d}{d x} [- 1]) + (x - 1) \frac{d}{d x} [e^{x}]) - e^{x} (x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.5.2

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

$3 \frac{x^{2} (e^{x} (1 + \frac{d}{d x} [- 1]) + (x - 1) \frac{d}{d x} [e^{x}]) - e^{x} (x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.5.3

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

$3 \frac{x^{2} (e^{x} (1 + 0) + (x - 1) \frac{d}{d x} [e^{x}]) - e^{x} (x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.5.4

Simplify the expression.

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Step 2.5.4.1

Add $1$ and $0$ .

$3 \frac{x^{2} (e^{x} \cdot 1 + (x - 1) \frac{d}{d x} [e^{x}]) - e^{x} (x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.5.4.2

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

$3 \frac{x^{2} (e^{x} + (x - 1) \frac{d}{d x} [e^{x}]) - e^{x} (x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.6

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

$3 \frac{x^{2} (e^{x} + (x - 1) e^{x}) - e^{x} (x - 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.7

Differentiate using the Power Rule.

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Step 2.7.1

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

$3 \frac{x^{2} (e^{x} + (x - 1) e^{x}) - e^{x} (x - 1) (2 x)}{x^{4}}$

Step 2.7.2

Simplify with factoring out.

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Step 2.7.2.1

Multiply $2$ by $- 1$ .

$3 \frac{x^{2} (e^{x} + (x - 1) e^{x}) - 2 e^{x} (x - 1) x}{x^{4}}$

Step 2.7.2.2

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

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Step 2.7.2.2.1

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

$3 \frac{x (x (e^{x} + (x - 1) e^{x})) - 2 e^{x} (x - 1) x}{x^{4}}$

Step 2.7.2.2.2

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

$3 \frac{x (x (e^{x} + (x - 1) e^{x})) + x (- 2 e^{x} (x - 1))}{x^{4}}$

Step 2.7.2.2.3

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

$3 \frac{x (x (e^{x} + (x - 1) e^{x}) - 2 e^{x} (x - 1))}{x^{4}}$

Step 2.8

Cancel the common factors.

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Step 2.8.1

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

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

Step 2.8.2

Cancel the common factor.

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

Step 2.8.3

Rewrite the expression.

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

Step 2.9

Combine $3$ and $\frac{x (e^{x} + (x - 1) e^{x}) - 2 e^{x} (x - 1)}{x^{3}}$ .

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

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

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

Step 2.10.2

Apply the distributive property.

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

Step 2.10.3

Apply the distributive property.

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

Step 2.10.4

Apply the distributive property.

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

Step 2.10.5

Simplify the numerator.

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Step 2.10.5.1

Simplify each term.

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Step 2.10.5.1.1

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

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Step 2.10.5.1.1.1

Move $x$ .

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

Step 2.10.5.1.1.2

Multiply $x$ by $x$ .

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

Step 2.10.5.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.10.5.1.3

Multiply $- 1$ by $3$ .

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

Step 2.10.5.1.4

Multiply $- 2$ by $3$ .

$\frac{3 x e^{x} + 3 x^{2} e^{x} - 3 x e^{x} - 6 (e^{x} x) + 3 (- 2 e^{x} \cdot - 1)}{x^{3}}$

Step 2.10.5.1.5

Multiply $- 1$ by $- 2$ .

$\frac{3 x e^{x} + 3 x^{2} e^{x} - 3 x e^{x} - 6 e^{x} x + 3 (2 e^{x})}{x^{3}}$

Step 2.10.5.1.6

Multiply $2$ by $3$ .

$\frac{3 x e^{x} + 3 x^{2} e^{x} - 3 x e^{x} - 6 e^{x} x + 6 e^{x}}{x^{3}}$

Step 2.10.5.2

Combine the opposite terms in $3 x e^{x} + 3 x^{2} e^{x} - 3 x e^{x} - 6 e^{x} x + 6 e^{x}$ .

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Step 2.10.5.2.1

Subtract $3 x e^{x}$ from $3 x e^{x}$ .

$\frac{3 x^{2} e^{x} + 0 - 6 e^{x} x + 6 e^{x}}{x^{3}}$

Step 2.10.5.2.2

Add $3 x^{2} e^{x}$ and $0$ .

$\frac{3 x^{2} e^{x} - 6 e^{x} x + 6 e^{x}}{x^{3}}$

Step 2.10.5.3

Reorder factors in $3 x^{2} e^{x} - 6 e^{x} x + 6 e^{x}$ .

$\frac{3 x^{2} e^{x} - 6 x e^{x} + 6 e^{x}}{x^{3}}$

Step 2.10.6

Reorder terms.

$\frac{3 e^{x} x^{2} - 6 e^{x} x + 6 e^{x}}{x^{3}}$

Step 2.10.7

Factor $3 e^{x}$ out of $3 e^{x} x^{2} - 6 e^{x} x + 6 e^{x}$ .

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Step 2.10.7.1

Factor $3 e^{x}$ out of $3 e^{x} x^{2}$ .

$\frac{3 e^{x} (x^{2}) - 6 e^{x} x + 6 e^{x}}{x^{3}}$

Step 2.10.7.2

Factor $3 e^{x}$ out of $- 6 e^{x} x$ .

$\frac{3 e^{x} (x^{2}) + 3 e^{x} (- 2 x) + 6 e^{x}}{x^{3}}$

Step 2.10.7.3

Factor $3 e^{x}$ out of $6 e^{x}$ .

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

Step 2.10.7.4

Factor $3 e^{x}$ out of $3 e^{x} (x^{2}) + 3 e^{x} (- 2 x)$ .

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

Step 2.10.7.5

Factor $3 e^{x}$ out of $3 e^{x} (x^{2} - 2 x) + 3 e^{x} (2)$ .

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

Step 3

The second derivative of $g (x)$ with respect to $x$ is $\frac{3 e^{x} (x^{2} - 2 x + 2)}{x^{3}}$ .

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