Calculus Examples

$y = x^{4} + 9 e^{x}$ , $(0, 9)$

Step 1

Find the first derivative and evaluate at $x = 0$ and $y = 9$ to find the slope of the tangent line.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{4} + 9 e^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [9 e^{x}]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [9 e^{x}]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$4 x^{3} + \frac{d}{d x} [9 e^{x}]$

Step 1.2

Evaluate $\frac{d}{d x} [9 e^{x}]$ .

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Step 1.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 e^{x}$ with respect to $x$ is $9 \frac{d}{d x} [e^{x}]$ .

$4 x^{3} + 9 \frac{d}{d x} [e^{x}]$

Step 1.2.2

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

$4 x^{3} + 9 e^{x}$

Step 1.3

Evaluate the derivative at $x = 0$ .

$4 {(0)}^{3} + 9 e^{0}$

Step 1.4

Simplify.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Raising $0$ to any positive power yields $0$ .

$4 \cdot 0 + 9 e^{0}$

Step 1.4.1.2

Multiply $4$ by $0$ .

$0 + 9 e^{0}$

Step 1.4.1.3

Anything raised to $0$ is $1$ .

$0 + 9 \cdot 1$

Step 1.4.1.4

Multiply $9$ by $1$ .

$0 + 9$

Step 1.4.2

Add $0$ and $9$ .

$9$

Step 2

The normal line is perpendicular to the tangent line. Take the negative reciprocal of the slope of the tangent line to find the slope of the normal line.

$- \frac{1}{9}$

Step 3

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

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Step 3.1

Use the slope $- \frac{1}{9}$ and a given point $(0, 9)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (9) = - \frac{1}{9} \cdot (x - (0))$

Step 3.2

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

$y - 9 = - \frac{1}{9} \cdot (x + 0)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- \frac{1}{9} \cdot (x + 0)$ .

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Step 3.3.1.1

Add $x$ and $0$ .

$y - 9 = - \frac{1}{9} \cdot x$

Step 3.3.1.2

Combine $x$ and $\frac{1}{9}$ .

$y - 9 = - \frac{x}{9}$

Step 3.3.2

Add $9$ to both sides of the equation.

$y = - \frac{x}{9} + 9$

Step 3.3.3

Write in $y = m x + b$ form.

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Step 3.3.3.1

Reorder terms.

$y = - (\frac{1}{9} x) + 9$

Step 3.3.3.2

Remove parentheses.

$y = - \frac{1}{9} x + 9$

Step 4