Calculus Examples

$f (x) = {(6 x - 5)}^{\frac{1}{2}}$ , $x = 1$

Step 1

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

Tap for more steps...

Step 1.1

Substitute $1$ in for $x$ .

$y = {(6 (1) - 5)}^{\frac{1}{2}}$

Step 1.2

Solve for $y$ .

Tap for more steps...

Step 1.2.1

Remove parentheses.

$y = {(6 (1) - 5)}^{\frac{1}{2}}$

Step 1.2.2

Simplify ${(6 (1) - 5)}^{\frac{1}{2}}$ .

Tap for more steps...

Step 1.2.2.1

Multiply $6$ by $1$ .

$y = {(6 - 5)}^{\frac{1}{2}}$

Step 1.2.2.2

Subtract $5$ from $6$ .

$y = 1^{\frac{1}{2}}$

Step 1.2.2.3

One to any power is one.

$y = 1$

Step 2

Find the first derivative and evaluate at $x = 1$ and $y = 1$ 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 $f (x) = x^{\frac{1}{2}}$ and $g (x) = 6 x - 5$ .

Tap for more steps...

Step 2.1.1

To apply the Chain Rule, set $u$ as $6 x - 5$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [6 x - 5]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{2}$ .

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [6 x - 5]$

Step 2.1.3

Replace all occurrences of $u$ with $6 x - 5$ .

$\frac{1}{2} {(6 x - 5)}^{\frac{1}{2} - 1} \frac{d}{d x} [6 x - 5]$

Step 2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} {(6 x - 5)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [6 x - 5]$

Step 2.3

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

$\frac{1}{2} {(6 x - 5)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [6 x - 5]$

Step 2.4

Combine the numerators over the common denominator.

$\frac{1}{2} {(6 x - 5)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [6 x - 5]$

Step 2.5

Simplify the numerator.

Tap for more steps...

Step 2.5.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(6 x - 5)}^{\frac{1 - 2}{2}} \frac{d}{d x} [6 x - 5]$

Step 2.5.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(6 x - 5)}^{\frac{- 1}{2}} \frac{d}{d x} [6 x - 5]$

Step 2.6

Combine fractions.

Tap for more steps...

Step 2.6.1

Move the negative in front of the fraction.

$\frac{1}{2} {(6 x - 5)}^{- \frac{1}{2}} \frac{d}{d x} [6 x - 5]$

Step 2.6.2

Combine $\frac{1}{2}$ and ${(6 x - 5)}^{- \frac{1}{2}}$ .

$\frac{{(6 x - 5)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [6 x - 5]$

Step 2.6.3

Move ${(6 x - 5)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(6 x - 5)}^{\frac{1}{2}}} \frac{d}{d x} [6 x - 5]$

Step 2.7

By the Sum Rule, the derivative of $6 x - 5$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [- 5]$ .

$\frac{1}{2 {(6 x - 5)}^{\frac{1}{2}}} (\frac{d}{d x} [6 x] + \frac{d}{d x} [- 5])$

Step 2.8

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$\frac{1}{2 {(6 x - 5)}^{\frac{1}{2}}} (6 \frac{d}{d x} [x] + \frac{d}{d x} [- 5])$

Step 2.9

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

$\frac{1}{2 {(6 x - 5)}^{\frac{1}{2}}} (6 \cdot 1 + \frac{d}{d x} [- 5])$

Step 2.10

Multiply $6$ by $1$ .

$\frac{1}{2 {(6 x - 5)}^{\frac{1}{2}}} (6 + \frac{d}{d x} [- 5])$

Step 2.11

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$\frac{1}{2 {(6 x - 5)}^{\frac{1}{2}}} (6 + 0)$

Step 2.12

Simplify terms.

Tap for more steps...

Step 2.12.1

Add $6$ and $0$ .

$\frac{1}{2 {(6 x - 5)}^{\frac{1}{2}}} \cdot 6$

Step 2.12.2

Combine $\frac{1}{2 {(6 x - 5)}^{\frac{1}{2}}}$ and $6$ .

$\frac{6}{2 {(6 x - 5)}^{\frac{1}{2}}}$

Step 2.12.3

Factor $2$ out of $6$ .

$\frac{2 \cdot 3}{2 {(6 x - 5)}^{\frac{1}{2}}}$

Step 2.13

Cancel the common factors.

Tap for more steps...

Step 2.13.1

Factor $2$ out of $2 {(6 x - 5)}^{\frac{1}{2}}$ .

$\frac{2 \cdot 3}{2 ({(6 x - 5)}^{\frac{1}{2}})}$

Step 2.13.2

Cancel the common factor.

$\frac{2 \cdot 3}{2 {(6 x - 5)}^{\frac{1}{2}}}$

Step 2.13.3

Rewrite the expression.

$\frac{3}{{(6 x - 5)}^{\frac{1}{2}}}$

Step 2.14

Evaluate the derivative at $x = 1$ .

$\frac{3}{{(6 (1) - 5)}^{\frac{1}{2}}}$

Step 2.15

Simplify.

Tap for more steps...

Step 2.15.1

Simplify the denominator.

Tap for more steps...

Step 2.15.1.1

Multiply $6$ by $1$ .

$\frac{3}{{(6 - 5)}^{\frac{1}{2}}}$

Step 2.15.1.2

Subtract $5$ from $6$ .

$\frac{3}{1^{\frac{1}{2}}}$

Step 2.15.1.3

One to any power is one.

$\frac{3}{1}$

Step 2.15.2

Divide $3$ by $1$ .

$3$

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 $3$ and a given point $(1, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = 3 \cdot (x - (1))$

Step 3.2

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

$y - 1 = 3 \cdot (x - 1)$

Step 3.3

Solve for $y$ .

Tap for more steps...

Step 3.3.1

Simplify $3 \cdot (x - 1)$ .

Tap for more steps...

Step 3.3.1.1

Rewrite.

$y - 1 = 0 + 0 + 3 \cdot (x - 1)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 1 = 3 \cdot (x - 1)$

Step 3.3.1.3

Apply the distributive property.

$y - 1 = 3 x + 3 \cdot - 1$

Step 3.3.1.4

Multiply $3$ by $- 1$ .

$y - 1 = 3 x - 3$

Step 3.3.2

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

Tap for more steps...

Step 3.3.2.1

Add $1$ to both sides of the equation.

$y = 3 x - 3 + 1$

Step 3.3.2.2

Add $- 3$ and $1$ .

$y = 3 x - 2$

Step 4