Calculus Examples

$g (x) = e^{x^{5} - 6 x}$ , $(- 1, e^{5})$

Step 1

Find the first derivative and evaluate at $x = - 1$ and $y = e^{5}$ to find the slope of the tangent line.

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Step 1.1

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 $g (x) = x^{5} - 6 x$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $x^{5} - 6 x$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [x^{5} - 6 x]$

Step 1.1.2

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

$e^{u} \frac{d}{d x} [x^{5} - 6 x]$

Step 1.1.3

Replace all occurrences of $u$ with $x^{5} - 6 x$ .

$e^{x^{5} - 6 x} \frac{d}{d x} [x^{5} - 6 x]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $x^{5} - 6 x$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 6 x]$ .

$e^{x^{5} - 6 x} (\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 6 x])$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$e^{x^{5} - 6 x} (5 x^{4} + \frac{d}{d x} [- 6 x])$

Step 1.2.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$e^{x^{5} - 6 x} (5 x^{4} - 6 \frac{d}{d x} [x])$

Step 1.2.4

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

$e^{x^{5} - 6 x} (5 x^{4} - 6 \cdot 1)$

Step 1.2.5

Multiply $- 6$ by $1$ .

$e^{x^{5} - 6 x} (5 x^{4} - 6)$

Step 1.3

Evaluate the derivative at $x = - 1$ .

$e^{{(- 1)}^{5} - 6 \cdot - 1} (5 {(- 1)}^{4} - 6)$

Step 1.4

Simplify.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Raise $- 1$ to the power of $5$ .

$e^{- 1 - 6 \cdot - 1} (5 {(- 1)}^{4} - 6)$

Step 1.4.1.2

Multiply $- 6$ by $- 1$ .

$e^{- 1 + 6} (5 {(- 1)}^{4} - 6)$

Step 1.4.2

Add $- 1$ and $6$ .

$e^{5} (5 {(- 1)}^{4} - 6)$

Step 1.4.3

Simplify each term.

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Step 1.4.3.1

Raise $- 1$ to the power of $4$ .

$e^{5} (5 \cdot 1 - 6)$

Step 1.4.3.2

Multiply $5$ by $1$ .

$e^{5} (5 - 6)$

Step 1.4.4

Simplify the expression.

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Step 1.4.4.1

Subtract $6$ from $5$ .

$e^{5} \cdot - 1$

Step 1.4.4.2

Move $- 1$ to the left of $e^{5}$ .

$- 1 \cdot e^{5}$

Step 1.4.4.3

Rewrite $- 1 e^{5}$ as $- e^{5}$ .

$- e^{5}$

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 $- e^{5}$ and a given point $(- 1, e^{5})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (e^{5}) = - e^{5} \cdot (x - (- 1))$

Step 2.2

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

$y - e^{5} = - e^{5} \cdot (x + 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- e^{5} \cdot (x + 1)$ .

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Step 2.3.1.1

Rewrite.

$y - e^{5} = 0 + 0 - e^{5} \cdot (x + 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - e^{5} = - e^{5} \cdot (x + 1)$

Step 2.3.1.3

Apply the distributive property.

$y - e^{5} = - e^{5} x - e^{5} \cdot 1$

Step 2.3.1.4

Multiply $- 1$ by $1$ .

$y - e^{5} = - e^{5} x - e^{5}$

Step 2.3.2

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

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Step 2.3.2.1

Add $e^{5}$ to both sides of the equation.

$y = - e^{5} x - e^{5} + e^{5}$

Step 2.3.2.2

Combine the opposite terms in $- e^{5} x - e^{5} + e^{5}$ .

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Step 2.3.2.2.1

Add $- e^{5}$ and $e^{5}$ .

$y = - e^{5} x + 0$

Step 2.3.2.2.2

Add $- e^{5} x$ and $0$ .

$y = - e^{5} x$

Step 3