Calculus Examples

$\frac{x^{2} - x - 11}{e^{x}}$

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 $f (x) = x^{2} - x - 11$ and $g (x) = e^{x}$ .

$\frac{e^{x} \frac{d}{d x} [x^{2} - x - 11] - (x^{2} - x - 11) \frac{d}{d x} [e^{x}]}{{(e^{x})}^{2}}$

Step 2

Differentiate.

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Step 2.1

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

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Step 2.1.1

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

$\frac{e^{x} \frac{d}{d x} [x^{2} - x - 11] - (x^{2} - x - 11) \frac{d}{d x} [e^{x}]}{e^{x \cdot 2}}$

Step 2.1.2

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

$\frac{e^{x} \frac{d}{d x} [x^{2} - x - 11] - (x^{2} - x - 11) \frac{d}{d x} [e^{x}]}{e^{2 x}}$

Step 2.2

By the Sum Rule, the derivative of $x^{2} - x - 11$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 11]$ .

$\frac{e^{x} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 11]) - (x^{2} - x - 11) \frac{d}{d x} [e^{x}]}{e^{2 x}}$

Step 2.3

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

$\frac{e^{x} (2 x + \frac{d}{d x} [- x] + \frac{d}{d x} [- 11]) - (x^{2} - x - 11) \frac{d}{d x} [e^{x}]}{e^{2 x}}$

Step 2.4

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

$\frac{e^{x} (2 x - \frac{d}{d x} [x] + \frac{d}{d x} [- 11]) - (x^{2} - x - 11) \frac{d}{d x} [e^{x}]}{e^{2 x}}$

Step 2.5

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

$\frac{e^{x} (2 x - 1 \cdot 1 + \frac{d}{d x} [- 11]) - (x^{2} - x - 11) \frac{d}{d x} [e^{x}]}{e^{2 x}}$

Step 2.6

Multiply $- 1$ by $1$ .

$\frac{e^{x} (2 x - 1 + \frac{d}{d x} [- 11]) - (x^{2} - x - 11) \frac{d}{d x} [e^{x}]}{e^{2 x}}$

Step 2.7

Since $- 11$ is constant with respect to $x$ , the derivative of $- 11$ with respect to $x$ is $0$ .

$\frac{e^{x} (2 x - 1 + 0) - (x^{2} - x - 11) \frac{d}{d x} [e^{x}]}{e^{2 x}}$

Step 2.8

Add $2 x - 1$ and $0$ .

$\frac{e^{x} (2 x - 1) - (x^{2} - x - 11) \frac{d}{d x} [e^{x}]}{e^{2 x}}$

Step 3

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

$\frac{e^{x} (2 x - 1) - (x^{2} - x - 11) e^{x}}{e^{2 x}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{e^{x} (2 x) + e^{x} \cdot - 1 - (x^{2} - x - 11) e^{x}}{e^{2 x}}$

Step 4.2

Apply the distributive property.

$\frac{e^{x} (2 x) + e^{x} \cdot - 1 + (- x^{2} - - x - - 11) e^{x}}{e^{2 x}}$

Step 4.3

Apply the distributive property.

$\frac{e^{x} (2 x) + e^{x} \cdot - 1 - x^{2} e^{x} - - x e^{x} - - 11 e^{x}}{e^{2 x}}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Simplify each term.

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Step 4.4.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 e^{x} x + e^{x} \cdot - 1 - x^{2} e^{x} - - x e^{x} - - 11 e^{x}}{e^{2 x}}$

Step 4.4.1.2

Move $- 1$ to the left of $e^{x}$ .

$\frac{2 e^{x} x - 1 \cdot e^{x} - x^{2} e^{x} - - x e^{x} - - 11 e^{x}}{e^{2 x}}$

Step 4.4.1.3

Rewrite $- 1 e^{x}$ as $- e^{x}$ .

$\frac{2 e^{x} x - e^{x} - x^{2} e^{x} - - x e^{x} - - 11 e^{x}}{e^{2 x}}$

Step 4.4.1.4

Multiply $- - x$ .

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Step 4.4.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{2 e^{x} x - e^{x} - x^{2} e^{x} + 1 x e^{x} - - 11 e^{x}}{e^{2 x}}$

Step 4.4.1.4.2

Multiply $x$ by $1$ .

$\frac{2 e^{x} x - e^{x} - x^{2} e^{x} + x e^{x} - - 11 e^{x}}{e^{2 x}}$

Step 4.4.1.5

Multiply $- 1$ by $- 11$ .

$\frac{2 e^{x} x - e^{x} - x^{2} e^{x} + x e^{x} + 11 e^{x}}{e^{2 x}}$

Step 4.4.2

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

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Step 4.4.2.1

Move $e^{x}$ .

$\frac{2 x e^{x} + x e^{x} - e^{x} - x^{2} e^{x} + 11 e^{x}}{e^{2 x}}$

Step 4.4.2.2

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

$\frac{3 x e^{x} - e^{x} - x^{2} e^{x} + 11 e^{x}}{e^{2 x}}$

Step 4.4.3

Add $- e^{x}$ and $11 e^{x}$ .

$\frac{3 x e^{x} - x^{2} e^{x} + 10 e^{x}}{e^{2 x}}$

Step 4.5

Reorder terms.

$\frac{3 e^{x} x - e^{x} x^{2} + 10 e^{x}}{e^{2 x}}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Factor $e^{x}$ out of $3 e^{x} x - e^{x} x^{2} + 10 e^{x}$ .

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Step 4.6.1.1

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

$\frac{e^{x} (3 x) - e^{x} x^{2} + 10 e^{x}}{e^{2 x}}$

Step 4.6.1.2

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

$\frac{e^{x} (3 x) + e^{x} (- x^{2}) + 10 e^{x}}{e^{2 x}}$

Step 4.6.1.3

Factor $e^{x}$ out of $10 e^{x}$ .

$\frac{e^{x} (3 x) + e^{x} (- x^{2}) + e^{x} \cdot 10}{e^{2 x}}$

Step 4.6.1.4

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

$\frac{e^{x} (3 x - x^{2}) + e^{x} \cdot 10}{e^{2 x}}$

Step 4.6.1.5

Factor $e^{x}$ out of $e^{x} (3 x - x^{2}) + e^{x} \cdot 10$ .

$\frac{e^{x} (3 x - x^{2} + 10)}{e^{2 x}}$

Step 4.6.2

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$\frac{e^{x} (3 u - u^{2} + 10)}{e^{2 x}}$

Step 4.6.3

Factor by grouping.

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Step 4.6.3.1

Reorder terms.

$\frac{e^{x} (- u^{2} + 3 u + 10)}{e^{2 x}}$

Step 4.6.3.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 10 = - 10$ and whose sum is $b = 3$ .

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Step 4.6.3.2.1

Factor $3$ out of $3 u$ .

$\frac{e^{x} (- u^{2} + 3 (u) + 10)}{e^{2 x}}$

Step 4.6.3.2.2

Rewrite $3$ as $- 2$ plus $5$

$\frac{e^{x} (- u^{2} + (- 2 + 5) u + 10)}{e^{2 x}}$

Step 4.6.3.2.3

Apply the distributive property.

$\frac{e^{x} (- u^{2} - 2 u + 5 u + 10)}{e^{2 x}}$

Step 4.6.3.3

Factor out the greatest common factor from each group.

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Step 4.6.3.3.1

Group the first two terms and the last two terms.

$\frac{e^{x} ((- u^{2} - 2 u) + 5 u + 10)}{e^{2 x}}$

Step 4.6.3.3.2

Factor out the greatest common factor (GCF) from each group.

$\frac{e^{x} (u (- u - 2) - 5 (- u - 2))}{e^{2 x}}$

Step 4.6.3.4

Factor the polynomial by factoring out the greatest common factor, $- u - 2$ .

$\frac{e^{x} ((- u - 2) (u - 5))}{e^{2 x}}$

Step 4.6.4

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

$\frac{e^{x} (- x - 2) (x - 5)}{e^{2 x}}$

Step 4.7

Cancel the common factor of $e^{x}$ and $e^{2 x}$ .

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Step 4.7.1

Factor $e^{2 x}$ out of $e^{x} (- x - 2) (x - 5)$ .

$\frac{e^{2 x} ((e^{- x} (- x - 2)) (x - 5))}{e^{2 x}}$

Step 4.7.2

Cancel the common factors.

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Step 4.7.2.1

Multiply by $1$ .

$\frac{e^{2 x} ((e^{- x} (- x - 2)) (x - 5))}{e^{2 x} \cdot 1}$

Step 4.7.2.2

Cancel the common factor.

$\frac{e^{2 x} ((e^{- x} (- x - 2)) (x - 5))}{e^{2 x} \cdot 1}$

Step 4.7.2.3

Rewrite the expression.

$\frac{(e^{- x} (- x - 2)) (x - 5)}{1}$

Step 4.7.2.4

Divide $(e^{- x} (- x - 2)) (x - 5)$ by $1$ .

$(e^{- x} (- x - 2)) (x - 5)$

Step 4.8

Apply the distributive property.

$(e^{- x} (- x) + e^{- x} \cdot - 2) (x - 5)$

Step 4.9

Rewrite using the commutative property of multiplication.

$(- e^{- x} x + e^{- x} \cdot - 2) (x - 5)$

Step 4.10

Move $- 2$ to the left of $e^{- x}$ .

$(- e^{- x} x - 2 \cdot e^{- x}) (x - 5)$

Step 4.11

Expand $(- e^{- x} x - 2 e^{- x}) (x - 5)$ using the FOIL Method.

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Step 4.11.1

Apply the distributive property.

$- e^{- x} x (x - 5) - 2 e^{- x} (x - 5)$

Step 4.11.2

Apply the distributive property.

$- e^{- x} x \cdot x - e^{- x} x \cdot - 5 - 2 e^{- x} (x - 5)$

Step 4.11.3

Apply the distributive property.

$- e^{- x} x \cdot x - e^{- x} x \cdot - 5 - 2 e^{- x} x - 2 e^{- x} \cdot - 5$

Step 4.12

Simplify and combine like terms.

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Step 4.12.1

Simplify each term.

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Step 4.12.1.1

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

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Step 4.12.1.1.1

Move $x$ .

$- e^{- x} (x \cdot x) - e^{- x} x \cdot - 5 - 2 e^{- x} x - 2 e^{- x} \cdot - 5$

Step 4.12.1.1.2

Multiply $x$ by $x$ .

$- e^{- x} x^{2} - e^{- x} x \cdot - 5 - 2 e^{- x} x - 2 e^{- x} \cdot - 5$

Step 4.12.1.2

Multiply $- 5$ by $- 1$ .

$- e^{- x} x^{2} + 5 e^{- x} x - 2 e^{- x} x - 2 e^{- x} \cdot - 5$

Step 4.12.1.3

Multiply $- 5$ by $- 2$ .

$- e^{- x} x^{2} + 5 e^{- x} x - 2 e^{- x} x + 10 e^{- x}$

Step 4.12.2

Subtract $2 e^{- x} x$ from $5 e^{- x} x$ .

$- e^{- x} x^{2} + 3 e^{- x} x + 10 e^{- x}$

Step 4.13

Reorder factors in $- e^{- x} x^{2} + 3 e^{- x} x + 10 e^{- x}$ .

$- x^{2} e^{- x} + 3 x e^{- x} + 10 e^{- x}$