Calculus Examples

$y = (x^{2} - 80) e^{x}$

Step 1

Apply the distributive property.

$y = x^{2} e^{x} - 80 e^{x}$

Step 2

Set $y$ as a function of $x$ .

$f (x) = x^{2} e^{x} - 80 e^{x}$

Step 3

Find the derivative.

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Step 3.1

By the Sum Rule, the derivative of $x^{2} e^{x} - 80 e^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{2} e^{x}] + \frac{d}{d x} [- 80 e^{x}]$ .

$\frac{d}{d x} [x^{2} e^{x}] + \frac{d}{d x} [- 80 e^{x}]$

Step 3.2

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

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Step 3.2.1

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

$x^{2} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 80 e^{x}]$

Step 3.2.2

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

$x^{2} e^{x} + e^{x} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 80 e^{x}]$

Step 3.2.3

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

$x^{2} e^{x} + e^{x} (2 x) + \frac{d}{d x} [- 80 e^{x}]$

Step 3.3

Evaluate $\frac{d}{d x} [- 80 e^{x}]$ .

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Step 3.3.1

Since $- 80$ is constant with respect to $x$ , the derivative of $- 80 e^{x}$ with respect to $x$ is $- 80 \frac{d}{d x} [e^{x}]$ .

$x^{2} e^{x} + e^{x} (2 x) - 80 \frac{d}{d x} [e^{x}]$

Step 3.3.2

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

$x^{2} e^{x} + e^{x} (2 x) - 80 e^{x}$

Step 3.4

Simplify.

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Step 3.4.1

Reorder terms.

$e^{x} x^{2} + 2 e^{x} x - 80 e^{x}$

Step 3.4.2

Reorder factors in $e^{x} x^{2} + 2 e^{x} x - 80 e^{x}$ .

$x^{2} e^{x} + 2 x e^{x} - 80 e^{x}$

Step 4

Set the derivative equal to $0$ then solve the equation $x^{2} e^{x} + 2 x e^{x} - 80 e^{x} = 0$ .

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Step 4.1

Factor the left side of the equation.

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Step 4.1.1

Factor $e^{x}$ out of $x^{2} e^{x} + 2 x e^{x} - 80 e^{x}$ .

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Step 4.1.1.1

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

$e^{x} x^{2} + 2 x e^{x} - 80 e^{x} = 0$

Step 4.1.1.2

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

$e^{x} x^{2} + e^{x} (2 x) - 80 e^{x} = 0$

Step 4.1.1.3

Factor $e^{x}$ out of $- 80 e^{x}$ .

$e^{x} x^{2} + e^{x} (2 x) + e^{x} \cdot - 80 = 0$

Step 4.1.1.4

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

$e^{x} (x^{2} + 2 x) + e^{x} \cdot - 80 = 0$

Step 4.1.1.5

Factor $e^{x}$ out of $e^{x} (x^{2} + 2 x) + e^{x} \cdot - 80$ .

$e^{x} (x^{2} + 2 x - 80) = 0$

Step 4.1.2

Factor.

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Step 4.1.2.1

Factor $x^{2} + 2 x - 80$ using the AC method.

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Step 4.1.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 80$ and whose sum is $2$ .

$- 8, 10$

Step 4.1.2.1.2

Write the factored form using these integers.

$e^{x} ((x - 8) (x + 10)) = 0$

Step 4.1.2.2

Remove unnecessary parentheses.

$e^{x} (x - 8) (x + 10) = 0$

Step 4.2

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

$e^{x} = 0$

$x - 8 = 0$

$x + 10 = 0$

Step 4.3

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

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Step 4.3.1

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

$e^{x} = 0$

Step 4.3.2

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

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Step 4.3.2.1

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

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

Step 4.3.2.2

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

Undefined

Step 4.3.2.3

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

No solution

Step 4.4

Set $x - 8$ equal to $0$ and solve for $x$ .

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Step 4.4.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 4.4.2

Add $8$ to both sides of the equation.

$x = 8$

Step 4.5

Set $x + 10$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $x + 10$ equal to $0$ .

$x + 10 = 0$

Step 4.5.2

Subtract $10$ from both sides of the equation.

$x = - 10$

Step 4.6

The final solution is all the values that make $e^{x} (x - 8) (x + 10) = 0$ true.

$x = 8, - 10$

Step 5

Solve the original function $f (x) = x^{2} e^{x} - 80 e^{x}$ at $x = 8$ .

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Step 5.1

Replace the variable $x$ with $8$ in the expression.

$f (8) = {(8)}^{2} e^{8} - 80 e^{8}$

Step 5.2

Simplify the result.

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Step 5.2.1

Raise $8$ to the power of $2$ .

$f (8) = 64 e^{8} - 80 e^{8}$

Step 5.2.2

Subtract $80 e^{8}$ from $64 e^{8}$ .

$f (8) = - 16 e^{8}$

Step 5.2.3

The final answer is $- 16 e^{8}$ .

$- 16 e^{8}$

Step 6

Solve the original function $f (x) = x^{2} e^{x} - 80 e^{x}$ at $x = - 10$ .

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Step 6.1

Replace the variable $x$ with $- 10$ in the expression.

$f (- 10) = {(- 10)}^{2} e^{- 10} - 80 e^{- 10}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $- 10$ to the power of $2$ .

$f (- 10) = 100 e^{- 10} - 80 e^{- 10}$

Step 6.2.1.2

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

$f (- 10) = 100 (\frac{1}{e^{10}}) - 80 e^{- 10}$

Step 6.2.1.3

Combine $100$ and $\frac{1}{e^{10}}$ .

$f (- 10) = \frac{100}{e^{10}} - 80 e^{- 10}$

Step 6.2.1.4

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

$f (- 10) = \frac{100}{e^{10}} - 80 \frac{1}{e^{10}}$

Step 6.2.1.5

Combine $- 80$ and $\frac{1}{e^{10}}$ .

$f (- 10) = \frac{100}{e^{10}} + \frac{- 80}{e^{10}}$

Step 6.2.1.6

Move the negative in front of the fraction.

$f (- 10) = \frac{100}{e^{10}} - \frac{80}{e^{10}}$

Step 6.2.2

Combine fractions.

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Step 6.2.2.1

Combine the numerators over the common denominator.

$f (- 10) = \frac{100 - 80}{e^{10}}$

Step 6.2.2.2

Subtract $80$ from $100$ .

$f (- 10) = \frac{20}{e^{10}}$

Step 6.2.3

The final answer is $\frac{20}{e^{10}}$ .

$\frac{20}{e^{10}}$

Step 7

The horizontal tangent lines on function $f (x) = x^{2} e^{x} - 80 e^{x}$ are $y = - 16 e^{8}, y = \frac{20}{e^{10}}$ .

$y = - 16 e^{8}, y = \frac{20}{e^{10}}$

Step 8