Calculus Examples

$\frac{\ln (x)}{\sqrt{x} + 6}$

Step 1

Find the $y$ value at $x = 1$ .

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Step 1.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \frac{\ln (1) (\sqrt{1} - 6)}{(1) - 36}$

Step 1.2

Simplify the result.

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Step 1.2.1

Simplify the numerator.

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Step 1.2.1.1

Any root of $1$ is $1$ .

$f (1) = \frac{\ln (1) (1 - 6)}{1 - 36}$

Step 1.2.1.2

Subtract $6$ from $1$ .

$f (1) = \frac{\ln (1) \cdot - 5}{1 - 36}$

Step 1.2.1.3

The natural logarithm of $1$ is $0$ .

$f (1) = \frac{0 \cdot - 5}{1 - 36}$

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

Subtract $36$ from $1$ .

$f (1) = \frac{0 \cdot - 5}{- 35}$

Step 1.2.2.2

Multiply $0$ by $- 5$ .

$f (1) = \frac{0}{- 35}$

Step 1.2.2.3

Divide $0$ by $- 35$ .

$f (1) = 0$

Step 1.2.3

The final answer is $0$ .

$0$

Step 1.3

The $y$ value at $x = 1$ is $0$ .

$y = 0$

Step 2

Find the $y$ value at $x = 2$ .

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Step 2.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = \frac{\ln (2) (\sqrt{2} - 6)}{(2) - 36}$

Step 2.2

Simplify the result.

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Step 2.2.1

Subtract $36$ from $2$ .

$f (2) = \frac{\ln (2) (\sqrt{2} - 6)}{- 34}$

Step 2.2.2

Move the negative in front of the fraction.

$f (2) = - \frac{\ln (2) (\sqrt{2} - 6)}{34}$

Step 2.2.3

The final answer is $- \frac{\ln (2) (\sqrt{2} - 6)}{34}$ .

$- \frac{\ln (2) (\sqrt{2} - 6)}{34}$

Step 2.3

The $y$ value at $x = 2$ is $- \frac{\ln (2) (\sqrt{2} - 6)}{34}$ .

$y = - \frac{\ln (2) (\sqrt{2} - 6)}{34}$

Step 3

Find the $y$ value at $x = 3$ .

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Step 3.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \frac{\ln (3) (\sqrt{3} - 6)}{(3) - 36}$

Step 3.2

Simplify the result.

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Step 3.2.1

Subtract $36$ from $3$ .

$f (3) = \frac{\ln (3) (\sqrt{3} - 6)}{- 33}$

Step 3.2.2

Move the negative in front of the fraction.

$f (3) = - \frac{\ln (3) (\sqrt{3} - 6)}{33}$

Step 3.2.3

The final answer is $- \frac{\ln (3) (\sqrt{3} - 6)}{33}$ .

$- \frac{\ln (3) (\sqrt{3} - 6)}{33}$

Step 3.3

The $y$ value at $x = 3$ is $- \frac{\ln (3) (\sqrt{3} - 6)}{33}$ .

$y = - \frac{\ln (3) (\sqrt{3} - 6)}{33}$

Step 4

List the points to graph.

$(1, 0), (2, 0.09348896), (3, 0.14208549)$

Step 5

Select a few points to graph.

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 0.093 \\ 3 & 0.142 \end{array}$

Step 6