Calculus Examples

$\ln (x)$ , $x = 1$

Step 1

Write $\ln (x)$ as an equation.

$y = \ln (x)$

Step 2

Find the corresponding $y$ -value to $x = 1$ .

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Step 2.1

Substitute $1$ in for $x$ .

$y = \ln (1)$

Step 2.2

Solve for $y$ .

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Step 2.2.1

Remove parentheses.

$y = \ln (1)$

Step 2.2.2

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

$y = 0$

Step 3

Find the first derivative and evaluate at $x = 1$ and $y = 0$ to find the slope of the tangent line.

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Step 3.1

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

$\frac{1}{x}$

Step 3.2

Evaluate the derivative at $x = 1$ .

$\frac{1}{1}$

Step 3.3

Divide $1$ by $1$ .

$1$

Step 4

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Solve for $y$ .

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Step 4.3.1

Add $y$ and $0$ .

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

Step 4.3.2

Multiply $x - 1$ by $1$ .

$y = x - 1$

Step 5