Calculus Examples

$f (x) = \frac{e^{x}}{x + 3}$ ,

$(0, \frac{1}{3})$

Step 1

Find the first derivative and evaluate at

$x = 0$ and

$y = \frac{1}{3}$ to find the slope of the tangent line.

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Step 1.1

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

$f (x) = e^{x}$ and

$g (x) = x + 3$ .

$\frac{(x + 3) \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [x + 3]}{{(x + 3)}^{2}}$

Step 1.2

Differentiate using the Exponential Rule which states that

$\frac{d}{d x} [a^{x}]$ is

$a^{x} \ln (a)$ where

$a$ =

$e$ .

$\frac{(x + 3) e^{x} - e^{x} \frac{d}{d x} [x + 3]}{{(x + 3)}^{2}}$

Step 1.3

Differentiate.

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Step 1.3.1

By the Sum Rule, the derivative of

$x + 3$ with respect to

$x$ is

$\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

$\frac{(x + 3) e^{x} - e^{x} (\frac{d}{d x} [x] + \frac{d}{d x} [3])}{{(x + 3)}^{2}}$

Step 1.3.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$\frac{(x + 3) e^{x} - e^{x} (1 + \frac{d}{d x} [3])}{{(x + 3)}^{2}}$

Step 1.3.3

Since

$3$ is constant with respect to

$x$ , the derivative of

$3$ with respect to

$x$ is

$0$ .

$\frac{(x + 3) e^{x} - e^{x} (1 + 0)}{{(x + 3)}^{2}}$

Step 1.3.4

Simplify the expression.

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Step 1.3.4.1

Add

$1$ and

$0$ .

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

Step 1.3.4.2

Multiply

$- 1$ by

$1$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Apply the distributive property.

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

Step 1.4.2

Subtract

$e^{x}$ from

$3 e^{x}$ .

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

Step 1.4.3

Reorder terms.

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

Step 1.4.4

Factor

$e^{x}$ out of

$e^{x} x + 2 e^{x}$ .

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Step 1.4.4.1

Factor

$e^{x}$ out of

$e^{x} x$ .

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

Step 1.4.4.2

Factor

$e^{x}$ out of

$2 e^{x}$ .

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

Step 1.4.4.3

Factor

$e^{x}$ out of

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

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

Step 1.5

Evaluate the derivative at

$x = 0$ .

$\frac{e^{0} ((0) + 2)}{{((0) + 3)}^{2}}$

Step 1.6

Simplify.

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Add

$0$ and

$2$ .

$\frac{e^{0} \cdot 2}{{(0 + 3)}^{2}}$

Step 1.6.1.2

Anything raised to

$0$ is

$1$ .

$\frac{1 \cdot 2}{{(0 + 3)}^{2}}$

Step 1.6.1.3

Multiply

$2$ by

$1$ .

$\frac{2}{{(0 + 3)}^{2}}$

Step 1.6.2

Simplify the denominator.

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Step 1.6.2.1

Add

$0$ and

$3$ .

$\frac{2}{3^{2}}$

Step 1.6.2.2

Raise

$3$ to the power of

$2$ .

$\frac{2}{9}$

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

$\frac{2}{9}$ and a given point

$(0, \frac{1}{3})$ 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

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{1}{3}) = \frac{2}{9} \cdot (x - (0))$

Step 2.2

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

$y - \frac{1}{3} = \frac{2}{9} \cdot (x + 0)$

Step 2.3

Solve for

$y$ .

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Step 2.3.1

Simplify

$\frac{2}{9} \cdot (x + 0)$ .

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Step 2.3.1.1

Add

$x$ and

$0$ .

$y - \frac{1}{3} = \frac{2}{9} \cdot x$

Step 2.3.1.2

Combine

$\frac{2}{9}$ and

$x$ .

$y - \frac{1}{3} = \frac{2 x}{9}$

Step 2.3.2

Add

$\frac{1}{3}$ to both sides of the equation.

$y = \frac{2 x}{9} + \frac{1}{3}$

Step 2.3.3

Reorder terms.

$y = \frac{2}{9} x + \frac{1}{3}$

Step 3