Calculus Examples

$y = x^{3} e^{- x}$ , $(1, \frac{1}{e})$

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = \frac{1}{e}$ to find the slope of the tangent line.

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Step 1.1

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 $f (x) = x^{3}$ and $g (x) = e^{- x}$ .

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

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - x$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $- x$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.3

Differentiate.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Simplify the expression.

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Step 1.3.3.1

Multiply $- 1$ by $1$ .

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

Step 1.3.3.2

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

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

Step 1.3.3.3

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

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

Step 1.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Reorder terms.

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

Step 1.4.2

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

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

Step 1.5

Evaluate the derivative at $x = 1$ .

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

Step 1.6

Simplify.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

One to any power is one.

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

Step 1.6.1.2

Multiply $- 1$ by $1$ .

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

Step 1.6.1.3

Multiply $- 1$ by $1$ .

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

Step 1.6.1.4

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

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

Step 1.6.1.5

Rewrite $- 1 \frac{1}{e}$ as $- \frac{1}{e}$ .

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

Step 1.6.1.6

One to any power is one.

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

Step 1.6.1.7

Multiply $3$ by $1$ .

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

Step 1.6.1.8

Multiply $- 1$ by $1$ .

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

Step 1.6.1.9

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

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

Step 1.6.1.10

Combine $3$ and $\frac{1}{e}$ .

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

Step 1.6.2

Combine fractions.

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Step 1.6.2.1

Combine the numerators over the common denominator.

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

Step 1.6.2.2

Add $- 1$ and $3$ .

$\frac{2}{e}$

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 2.1

Use the slope $\frac{2}{e}$ and a given point $(1, \frac{1}{e})$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{1}{e}) = \frac{2}{e} \cdot (x - (1))$

Step 2.2

Simplify the equation and keep it in point-slope form.

$y - \frac{1}{e} = \frac{2}{e} \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Move all terms containing variables to the left side of the equation.

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Step 2.3.1.1

Subtract $\frac{2}{e} \cdot (x - 1)$ from both sides of the equation.

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

Step 2.3.1.2

Simplify each term.

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Step 2.3.1.2.1

Apply the distributive property.

$y - \frac{1}{e} - \frac{2}{e} x - \frac{2}{e} \cdot - 1 = 0$

Step 2.3.1.2.2

Combine $x$ and $\frac{2}{e}$ .

$y - \frac{1}{e} - \frac{x \cdot 2}{e} - \frac{2}{e} \cdot - 1 = 0$

Step 2.3.1.2.3

Multiply $- \frac{2}{e} \cdot - 1$ .

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Step 2.3.1.2.3.1

Multiply $- 1$ by $- 1$ .

$y - \frac{1}{e} - \frac{x \cdot 2}{e} + 1 \frac{2}{e} = 0$

Step 2.3.1.2.3.2

Multiply $\frac{2}{e}$ by $1$ .

$y - \frac{1}{e} - \frac{x \cdot 2}{e} + \frac{2}{e} = 0$

Step 2.3.1.2.4

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

$y - \frac{1}{e} - \frac{2 x}{e} + \frac{2}{e} = 0$

Step 2.3.1.3

Combine the numerators over the common denominator.

$y + \frac{- 1 - 2 x + 2}{e} = 0$

Step 2.3.1.4

Add $- 1$ and $2$ .

$y + \frac{- 2 x + 1}{e} = 0$

Step 2.3.1.5

To write $y$ as a fraction with a common denominator, multiply by $\frac{e}{e}$ .

$\frac{y e}{e} + \frac{- 2 x + 1}{e} = 0$

Step 2.3.1.6

Combine the numerators over the common denominator.

$\frac{y e - 2 x + 1}{e} = 0$

Step 2.3.2

Set the numerator equal to zero.

$y e - 2 x + 1 = 0$

Step 2.3.3

Solve the equation for $y$ .

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Step 2.3.3.1

Move all terms not containing $y$ to the right side of the equation.

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Step 2.3.3.1.1

Add $2 x$ to both sides of the equation.

$y e + 1 = 2 x$

Step 2.3.3.1.2

Subtract $1$ from both sides of the equation.

$y e = 2 x - 1$

Step 2.3.3.2

Divide each term in $y e = 2 x - 1$ by $e$ and simplify.

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Step 2.3.3.2.1

Divide each term in $y e = 2 x - 1$ by $e$ .

$\frac{y e}{e} = \frac{2 x}{e} + \frac{- 1}{e}$

Step 2.3.3.2.2

Simplify the left side.

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Step 2.3.3.2.2.1

Cancel the common factor of $e$ .

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Step 2.3.3.2.2.1.1

Cancel the common factor.

$\frac{y e}{e} = \frac{2 x}{e} + \frac{- 1}{e}$

Step 2.3.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{2 x}{e} + \frac{- 1}{e}$

Step 2.3.3.2.3

Simplify the right side.

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Step 2.3.3.2.3.1

Move the negative in front of the fraction.

$y = \frac{2 x}{e} - \frac{1}{e}$

Step 2.3.4

Reorder terms.

$y = \frac{2}{e} x - \frac{1}{e}$

Step 3