Calculus Examples

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

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = e$ 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 $g (x) = x$ .

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

Step 1.2

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

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

Step 1.3

Differentiate using the Power Rule.

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Step 1.3.1

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

$\frac{x e^{x} - e^{x} \cdot 1}{x^{2}}$

Step 1.3.2

Multiply $- 1$ by $1$ .

$\frac{x e^{x} - e^{x}}{x^{2}}$

Step 1.4

Simplify.

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Step 1.4.1

Reorder terms.

$\frac{e^{x} x - e^{x}}{x^{2}}$

Step 1.4.2

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

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Step 1.4.2.1

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

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

Step 1.4.2.2

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

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

Step 1.4.2.3

Factor $e^{x}$ out of $e^{x} (x) + e^{x} \cdot - 1$ .

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

Step 1.5

Evaluate the derivative at $x = 1$ .

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

Step 1.6

Simplify.

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Subtract $1$ from $1$ .

$\frac{e^{1} \cdot 0}{1^{2}}$

Step 1.6.1.2

Simplify.

$\frac{e \cdot 0}{1^{2}}$

Step 1.6.2

Simplify the expression.

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Step 1.6.2.1

One to any power is one.

$\frac{e \cdot 0}{1}$

Step 1.6.2.2

Multiply $e$ by $0$ .

$\frac{0}{1}$

Step 1.6.2.3

Divide $0$ by $1$ .

$0$

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

$y - (e) = 0 \cdot (x - (1))$

Step 2.2

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

$y - e = 0 \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Multiply $0$ by $x - 1$ .

$y - e = 0$

Step 2.3.2

Add $e$ to both sides of the equation.

$y = e$

Step 3