Calculus Examples

$f (x) = 8 - \ln (x)$ ; $x = 1$

Step 1

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

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Step 1.1

Substitute $1$ in for $x$ .

$y = 8 - \ln (1)$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = 8 - \ln (1)$

Step 1.2.2

Simplify $8 - \ln (1)$ .

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

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

$y = 8 - 0$

Step 1.2.2.1.2

Multiply $- 1$ by $0$ .

$y = 8 + 0$

Step 1.2.2.2

Add $8$ and $0$ .

$y = 8$

Step 2

Find the first derivative and evaluate at $x = 1$ and $y = 8$ 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 $8 - \ln (x)$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [- \ln (x)]$ .

$\frac{d}{d x} [8] + \frac{d}{d x} [- \ln (x)]$

Step 2.1.2

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

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

Step 2.2

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

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Step 2.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \ln (x)$ with respect to $x$ is $- \frac{d}{d x} [\ln (x)]$ .

$0 - \frac{d}{d x} [\ln (x)]$

Step 2.2.2

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

$0 - \frac{1}{x}$

Step 2.3

Subtract $\frac{1}{x}$ from $0$ .

$- \frac{1}{x}$

Step 2.4

Evaluate the derivative at $x = 1$ .

$- \frac{1}{1}$

Step 2.5

Simplify.

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Step 2.5.1

Cancel the common factor of $1$ .

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Step 2.5.1.1

Cancel the common factor.

$- \frac{1}{1}$

Step 2.5.1.2

Rewrite the expression.

$- 1 \cdot 1$

Step 2.5.2

Multiply $- 1$ by $1$ .

$- 1$

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 $- 1$ and a given point $(1, 8)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (8) = - 1 \cdot (x - (1))$

Step 3.2

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

$y - 8 = - 1 \cdot (x - 1)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- 1 \cdot (x - 1)$ .

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Step 3.3.1.1

Rewrite.

$y - 8 = 0 + 0 - 1 \cdot (x - 1)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 8 = - 1 \cdot (x - 1)$

Step 3.3.1.3

Apply the distributive property.

$y - 8 = - 1 x - 1 \cdot - 1$

Step 3.3.1.4

Simplify the expression.

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Step 3.3.1.4.1

Rewrite $- 1 x$ as $- x$ .

$y - 8 = - x - 1 \cdot - 1$

Step 3.3.1.4.2

Multiply $- 1$ by $- 1$ .

$y - 8 = - x + 1$

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 $8$ to both sides of the equation.

$y = - x + 1 + 8$

Step 3.3.2.2

Add $1$ and $8$ .

$y = - x + 9$

Step 4