Calculus Examples

$y = \frac{e^{6 x}}{x}$ , $(\frac{1}{6}, 6 e)$

Step 1

Find the first derivative and evaluate at $x = \frac{1}{6}$ and $y = 6 e$ 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 $f (x) = e^{6 x}$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [e^{6 x}] - e^{6 x} \frac{d}{d x} [x]}{x^{2}}$

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 $g (x) = 6 x$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $6 x$ .

$\frac{x (\frac{d}{d u} [e^{u}] \frac{d}{d x} [6 x]) - e^{6 x} \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.2

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

$\frac{x (e^{u} \frac{d}{d x} [6 x]) - e^{6 x} \frac{d}{d x} [x]}{x^{2}}$

Step 1.2.3

Replace all occurrences of $u$ with $6 x$ .

$\frac{x (e^{6 x} \frac{d}{d x} [6 x]) - e^{6 x} \frac{d}{d x} [x]}{x^{2}}$

Step 1.3

Differentiate.

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Step 1.3.1

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

$\frac{x (e^{6 x} (6 \frac{d}{d x} [x])) - e^{6 x} \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.2

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

$\frac{x (e^{6 x} (6 \cdot 1)) - e^{6 x} \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.3

Simplify the expression.

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Step 1.3.3.1

Multiply $6$ by $1$ .

$\frac{x (e^{6 x} \cdot 6) - e^{6 x} \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.3.2

Move $6$ to the left of $e^{6 x}$ .

$\frac{x (6 \cdot e^{6 x}) - e^{6 x} \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.4

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

$\frac{x (6 e^{6 x}) - e^{6 x} \cdot 1}{x^{2}}$

Step 1.3.5

Multiply $- 1$ by $1$ .

$\frac{x (6 e^{6 x}) - e^{6 x}}{x^{2}}$

Step 1.4

Simplify.

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Step 1.4.1

Rewrite using the commutative property of multiplication.

$\frac{6 x e^{6 x} - e^{6 x}}{x^{2}}$

Step 1.4.2

Reorder terms.

$\frac{6 e^{6 x} x - e^{6 x}}{x^{2}}$

Step 1.4.3

Factor $e^{6 x}$ out of $6 e^{6 x} x - e^{6 x}$ .

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Step 1.4.3.1

Factor $e^{6 x}$ out of $6 e^{6 x} x$ .

$\frac{e^{6 x} (6 x) - e^{6 x}}{x^{2}}$

Step 1.4.3.2

Factor $e^{6 x}$ out of $- e^{6 x}$ .

$\frac{e^{6 x} (6 x) + e^{6 x} \cdot - 1}{x^{2}}$

Step 1.4.3.3

Factor $e^{6 x}$ out of $e^{6 x} (6 x) + e^{6 x} \cdot - 1$ .

$\frac{e^{6 x} (6 x - 1)}{x^{2}}$

Step 1.5

Evaluate the derivative at $x = \frac{1}{6}$ .

$\frac{e^{6 (\frac{1}{6})} (6 (\frac{1}{6}) - 1)}{{(\frac{1}{6})}^{2}}$

Step 1.6

Simplify.

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Cancel the common factor of $6$ .

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Step 1.6.1.1.1

Cancel the common factor.

$\frac{e^{6 (\frac{1}{6})} (6 (\frac{1}{6}) - 1)}{{(\frac{1}{6})}^{2}}$

Step 1.6.1.1.2

Rewrite the expression.

$\frac{e^{6 (\frac{1}{6})} (1 - 1)}{{(\frac{1}{6})}^{2}}$

Step 1.6.1.2

Subtract $1$ from $1$ .

$\frac{e^{6 (\frac{1}{6})} \cdot 0}{{(\frac{1}{6})}^{2}}$

Step 1.6.1.3

Cancel the common factor of $6$ .

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Step 1.6.1.3.1

Cancel the common factor.

$\frac{e^{6 (\frac{1}{6})} \cdot 0}{{(\frac{1}{6})}^{2}}$

Step 1.6.1.3.2

Rewrite the expression.

$\frac{e^{1} \cdot 0}{{(\frac{1}{6})}^{2}}$

Step 1.6.1.4

Simplify.

$\frac{e \cdot 0}{{(\frac{1}{6})}^{2}}$

Step 1.6.2

Simplify the denominator.

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Step 1.6.2.1

Apply the product rule to $\frac{1}{6}$ .

$\frac{e \cdot 0}{\frac{1^{2}}{6^{2}}}$

Step 1.6.2.2

One to any power is one.

$\frac{e \cdot 0}{\frac{1}{6^{2}}}$

Step 1.6.2.3

Raise $6$ to the power of $2$ .

$\frac{e \cdot 0}{\frac{1}{36}}$

Step 1.6.3

Multiply $e$ by $0$ .

$\frac{0}{\frac{1}{36}}$

Step 1.6.4

Multiply the numerator by the reciprocal of the denominator.

$0 \cdot 36$

Step 1.6.5

Multiply $0$ by $36$ .

$0$

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 $(\frac{1}{6}, 6 e)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (6 e) = 0 \cdot (x - (\frac{1}{6}))$

Step 2.2

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

$y - 6 e = 0 \cdot (x - \frac{1}{6})$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Multiply $0$ by $x - \frac{1}{6}$ .

$y - 6 e = 0$

Step 2.3.2

Add $6 e$ to both sides of the equation.

$y = 6 e$

Step 3