Calculus Examples

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

Step 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 $f (x) = e^{x} - 1 - x$ and $g (x) = x^{2}$ .

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

Step 2

Differentiate using the Sum Rule.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Multiply $2$ by $2$ .

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

Step 2.2

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

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

Step 3

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

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 4.4

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

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

Step 4.5

Multiply $- 1$ by $1$ .

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

Step 4.6

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

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

Step 4.7

Simplify with factoring out.

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Step 4.7.1

Multiply $2$ by $- 1$ .

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

Step 4.7.2

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

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Step 4.7.2.1

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

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

Step 4.7.2.2

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

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

Step 4.7.2.3

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

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

Step 5

Cancel the common factors.

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Step 5.1

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

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Simplify the numerator.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Move $- 1$ to the left of $x$ .

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

Step 6.3.1.2

Rewrite $- 1 x$ as $- x$ .

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

Step 6.3.1.3

Multiply $- 2$ by $- 1$ .

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

Step 6.3.1.4

Multiply $- 1$ by $- 2$ .

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

Step 6.3.2

Add $- x$ and $2 x$ .

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

Step 6.4

Reorder terms.

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

Step 6.5

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

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