Calculus Examples

$f (x) = \ln^{4} (x)$ at $x = 8$

Step 1

Find the corresponding $y$ -value to $x = 8$ .

Tap for more steps...

Step 1.1

Substitute $8$ in for $x$ .

$y = \ln ({(8)}^{4})$

Step 1.2

Solve for $y$ .

Tap for more steps...

Step 1.2.1

Remove parentheses.

$y = \ln (8^{4})$

Step 1.2.2

Remove parentheses.

$y = \ln ({(8)}^{4})$

Step 1.2.3

Raise $8$ to the power of $4$ .

$y = \ln (4096)$

Step 2

Find the first derivative and evaluate at $x = 8$ and $y = \ln (4096)$ to find the slope of the tangent line.

Tap for more steps...

Step 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 $g (x) = {(x)}^{4}$ .

Tap for more steps...

Step 2.1.1

To apply the Chain Rule, set $u$ as ${(x)}^{4}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [{(x)}^{4}]$

Step 2.1.2

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

$\frac{1}{u} \frac{d}{d x} [{(x)}^{4}]$

Step 2.1.3

Replace all occurrences of $u$ with ${(x)}^{4}$ .

$\frac{1}{x^{4}} \frac{d}{d x} [x^{4}]$

Step 2.2

Differentiate using the Power Rule.

Tap for more steps...

Step 2.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{1}{x^{4}} (4 x^{3})$

Step 2.2.2

Simplify terms.

Tap for more steps...

Step 2.2.2.1

Combine $4$ and $\frac{1}{x^{4}}$ .

$\frac{4}{x^{4}} x^{3}$

Step 2.2.2.2

Combine $\frac{4}{x^{4}}$ and $x^{3}$ .

$\frac{4 x^{3}}{x^{4}}$

Step 2.2.2.3

Cancel the common factor of $x^{3}$ and $x^{4}$ .

Tap for more steps...

Step 2.2.2.3.1

Factor $x^{3}$ out of $4 x^{3}$ .

$\frac{x^{3} \cdot 4}{x^{4}}$

Step 2.2.2.3.2

Cancel the common factors.

Tap for more steps...

Step 2.2.2.3.2.1

Factor $x^{3}$ out of $x^{4}$ .

$\frac{x^{3} \cdot 4}{x^{3} x}$

Step 2.2.2.3.2.2

Cancel the common factor.

$\frac{x^{3} \cdot 4}{x^{3} x}$

Step 2.2.2.3.2.3

Rewrite the expression.

$\frac{4}{x}$

Step 2.3

Evaluate the derivative at $x = 8$ .

$\frac{4}{8}$

Step 2.4

Cancel the common factor of $4$ and $8$ .

Tap for more steps...

Step 2.4.1

Factor $4$ out of $4$ .

$\frac{4 (1)}{8}$

Step 2.4.2

Cancel the common factors.

Tap for more steps...

Step 2.4.2.1

Factor $4$ out of $8$ .

$\frac{4 \cdot 1}{4 \cdot 2}$

Step 2.4.2.2

Cancel the common factor.

$\frac{4 \cdot 1}{4 \cdot 2}$

Step 2.4.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 3

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

Tap for more steps...

Step 3.1

Use the slope $\frac{1}{2}$ and a given point $(8, \ln (4096))$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\ln (4096)) = \frac{1}{2} \cdot (x - (8))$

Step 3.2

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

$y - \ln (4096) = \frac{1}{2} \cdot (x - 8)$

Step 3.3

Solve for $y$ .

Tap for more steps...

Step 3.3.1

Simplify $\frac{1}{2} \cdot (x - 8)$ .

Tap for more steps...

Step 3.3.1.1

Rewrite.

$y - \ln (4096) = 0 + 0 + \frac{1}{2} \cdot (x - 8)$

Step 3.3.1.2

Simplify by adding zeros.

$y - \ln (4096) = \frac{1}{2} \cdot (x - 8)$

Step 3.3.1.3

Apply the distributive property.

$y - \ln (4096) = \frac{1}{2} x + \frac{1}{2} \cdot - 8$

Step 3.3.1.4

Combine $\frac{1}{2}$ and $x$ .

$y - \ln (4096) = \frac{x}{2} + \frac{1}{2} \cdot - 8$

Step 3.3.1.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.1.5.1

Factor $2$ out of $- 8$ .

$y - \ln (4096) = \frac{x}{2} + \frac{1}{2} \cdot (2 (- 4))$

Step 3.3.1.5.2

Cancel the common factor.

$y - \ln (4096) = \frac{x}{2} + \frac{1}{2} \cdot (2 \cdot - 4)$

Step 3.3.1.5.3

Rewrite the expression.

$y - \ln (4096) = \frac{x}{2} - 4$

Step 3.3.2

Add $\ln (4096)$ to both sides of the equation.

$y = \frac{x}{2} - 4 + \ln (4096)$

Step 3.3.3

Reorder terms.

$y = \frac{1}{2} x - 4 + \ln (4096)$

Step 4