Calculus Examples

$f (x) = 2 + \ln (x^{5})$ ; $x = e$

Step 1

Find the corresponding $y$ -value to $x = e$ .

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Step 1.1

Substitute $e$ in for $x$ .

$y = 2 + \ln ({(e)}^{5})$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = 2 + \ln (e^{5})$

Step 1.2.2

Remove parentheses.

$y = 2 + \ln ({(e)}^{5})$

Step 1.2.3

Simplify $2 + \ln ({(e)}^{5})$ .

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Use logarithm rules to move $5$ out of the exponent.

$y = 2 + 5 \ln (e)$

Step 1.2.3.1.2

The natural logarithm of $e$ is $1$ .

$y = 2 + 5 \cdot 1$

Step 1.2.3.1.3

Multiply $5$ by $1$ .

$y = 2 + 5$

Step 1.2.3.2

Add $2$ and $5$ .

$y = 7$

Step 2

Find the first derivative and evaluate at $x = e$ and $y = 7$ to find the slope of the tangent line.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $2 + \ln (x^{5})$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [\ln (x^{5})]$ .

$\frac{d}{d x} [2] + \frac{d}{d x} [\ln (x^{5})]$

Step 2.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [\ln (x^{5})]$

Step 2.2

Evaluate $\frac{d}{d x} [\ln (x^{5})]$ .

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Step 2.2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x^{5}$ .

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Step 2.2.1.1

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

$0 + \frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{5}]$

Step 2.2.1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$0 + \frac{1}{u} \frac{d}{d x} [x^{5}]$

Step 2.2.1.3

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

$0 + \frac{1}{x^{5}} \frac{d}{d x} [x^{5}]$

Step 2.2.2

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

$0 + \frac{1}{x^{5}} (5 x^{4})$

Step 2.2.3

Combine $5$ and $\frac{1}{x^{5}}$ .

$0 + \frac{5}{x^{5}} x^{4}$

Step 2.2.4

Combine $\frac{5}{x^{5}}$ and $x^{4}$ .

$0 + \frac{5 x^{4}}{x^{5}}$

Step 2.2.5

Cancel the common factor of $x^{4}$ and $x^{5}$ .

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Step 2.2.5.1

Factor $x^{4}$ out of $5 x^{4}$ .

$0 + \frac{x^{4} \cdot 5}{x^{5}}$

Step 2.2.5.2

Cancel the common factors.

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Step 2.2.5.2.1

Factor $x^{4}$ out of $x^{5}$ .

$0 + \frac{x^{4} \cdot 5}{x^{4} x}$

Step 2.2.5.2.2

Cancel the common factor.

$0 + \frac{x^{4} \cdot 5}{x^{4} x}$

Step 2.2.5.2.3

Rewrite the expression.

$0 + \frac{5}{x}$

Step 2.3

Add $0$ and $\frac{5}{x}$ .

$\frac{5}{x}$

Step 2.4

Evaluate the derivative at $x = e$ .

$\frac{5}{e}$

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 $\frac{5}{e}$ and a given point $(e, 7)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (7) = \frac{5}{e} \cdot (x - (e))$

Step 3.2

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

$y - 7 = \frac{5}{e} \cdot (x - e)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $\frac{5}{e} \cdot (x - e)$ .

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Step 3.3.1.1

Rewrite.

$y - 7 = 0 + 0 + \frac{5}{e} \cdot (x - e)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 7 = \frac{5}{e} \cdot (x - e)$

Step 3.3.1.3

Apply the distributive property.

$y - 7 = \frac{5}{e} x + \frac{5}{e} (- e)$

Step 3.3.1.4

Combine $\frac{5}{e}$ and $x$ .

$y - 7 = \frac{5 x}{e} + \frac{5}{e} (- e)$

Step 3.3.1.5

Cancel the common factor of $e$ .

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Step 3.3.1.5.1

Factor $e$ out of $- e$ .

$y - 7 = \frac{5 x}{e} + \frac{5}{e} (e \cdot - 1)$

Step 3.3.1.5.2

Cancel the common factor.

$y - 7 = \frac{5 x}{e} + \frac{5}{e} (e \cdot - 1)$

Step 3.3.1.5.3

Rewrite the expression.

$y - 7 = \frac{5 x}{e} + 5 \cdot - 1$

Step 3.3.1.6

Multiply $5$ by $- 1$ .

$y - 7 = \frac{5 x}{e} - 5$

Step 3.3.2

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

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Step 3.3.2.1

Add $7$ to both sides of the equation.

$y = \frac{5 x}{e} - 5 + 7$

Step 3.3.2.2

Add $- 5$ and $7$ .

$y = \frac{5 x}{e} + 2$

Step 3.3.3

Reorder terms.

$y = \frac{5}{e} x + 2$

Step 4