Calculus Examples

${(12 x - 3)}^{\frac{1}{2}}$ $x = 1$

Step 1

Write ${(12 x - 3)}^{\frac{1}{2}}$ as an equation.

$y = {(12 x - 3)}^{\frac{1}{2}}$

Step 2

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

Tap for more steps...

Step 2.1

Substitute $1$ in for $x$ .

$y = {(12 (1) - 3)}^{\frac{1}{2}}$

Step 2.2

Solve for $y$ .

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = {(12 (1) - 3)}^{\frac{1}{2}}$

Step 2.2.2

Simplify ${(12 (1) - 3)}^{\frac{1}{2}}$ .

Tap for more steps...

Step 2.2.2.1

Simplify the expression.

Tap for more steps...

Step 2.2.2.1.1

Multiply $12$ by $1$ .

$y = {(12 - 3)}^{\frac{1}{2}}$

Step 2.2.2.1.2

Subtract $3$ from $12$ .

$y = 9^{\frac{1}{2}}$

Step 2.2.2.1.3

Rewrite $9$ as $3^{2}$ .

$y = {(3^{2})}^{\frac{1}{2}}$

Step 2.2.2.1.4

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = 3^{2 (\frac{1}{2})}$

Step 2.2.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.2.2.1

Cancel the common factor.

$y = 3^{2 (\frac{1}{2})}$

Step 2.2.2.2.2

Rewrite the expression.

$y = 3^{1}$

Step 2.2.2.3

Evaluate the exponent.

$y = 3$

Step 3

Find the first derivative and evaluate at $x = 1$ and $y = 3$ to find the slope of the tangent line.

Tap for more steps...

Step 3.1

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 $g (x) = 12 x - 3$ .

Tap for more steps...

Step 3.1.1

To apply the Chain Rule, set $u$ as $12 x - 3$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [12 x - 3]$

Step 3.1.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [12 x - 3]$

Step 3.1.3

Replace all occurrences of $u$ with $12 x - 3$ .

$\frac{1}{2} {(12 x - 3)}^{\frac{1}{2} - 1} \frac{d}{d x} [12 x - 3]$

Step 3.2

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

$\frac{1}{2} {(12 x - 3)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [12 x - 3]$

Step 3.3

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

$\frac{1}{2} {(12 x - 3)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [12 x - 3]$

Step 3.4

Combine the numerators over the common denominator.

$\frac{1}{2} {(12 x - 3)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [12 x - 3]$

Step 3.5

Simplify the numerator.

Tap for more steps...

Step 3.5.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(12 x - 3)}^{\frac{1 - 2}{2}} \frac{d}{d x} [12 x - 3]$

Step 3.5.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(12 x - 3)}^{\frac{- 1}{2}} \frac{d}{d x} [12 x - 3]$

Step 3.6

Combine fractions.

Tap for more steps...

Step 3.6.1

Move the negative in front of the fraction.

$\frac{1}{2} {(12 x - 3)}^{- \frac{1}{2}} \frac{d}{d x} [12 x - 3]$

Step 3.6.2

Combine $\frac{1}{2}$ and ${(12 x - 3)}^{- \frac{1}{2}}$ .

$\frac{{(12 x - 3)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [12 x - 3]$

Step 3.6.3

Move ${(12 x - 3)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(12 x - 3)}^{\frac{1}{2}}} \frac{d}{d x} [12 x - 3]$

Step 3.7

By the Sum Rule, the derivative of $12 x - 3$ with respect to $x$ is $\frac{d}{d x} [12 x] + \frac{d}{d x} [- 3]$ .

$\frac{1}{2 {(12 x - 3)}^{\frac{1}{2}}} (\frac{d}{d x} [12 x] + \frac{d}{d x} [- 3])$

Step 3.8

Since $12$ is constant with respect to $x$ , the derivative of $12 x$ with respect to $x$ is $12 \frac{d}{d x} [x]$ .

$\frac{1}{2 {(12 x - 3)}^{\frac{1}{2}}} (12 \frac{d}{d x} [x] + \frac{d}{d x} [- 3])$

Step 3.9

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

$\frac{1}{2 {(12 x - 3)}^{\frac{1}{2}}} (12 \cdot 1 + \frac{d}{d x} [- 3])$

Step 3.10

Multiply $12$ by $1$ .

$\frac{1}{2 {(12 x - 3)}^{\frac{1}{2}}} (12 + \frac{d}{d x} [- 3])$

Step 3.11

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$\frac{1}{2 {(12 x - 3)}^{\frac{1}{2}}} (12 + 0)$

Step 3.12

Simplify terms.

Tap for more steps...

Step 3.12.1

Add $12$ and $0$ .

$\frac{1}{2 {(12 x - 3)}^{\frac{1}{2}}} \cdot 12$

Step 3.12.2

Combine $\frac{1}{2 {(12 x - 3)}^{\frac{1}{2}}}$ and $12$ .

$\frac{12}{2 {(12 x - 3)}^{\frac{1}{2}}}$

Step 3.12.3

Factor $2$ out of $12$ .

$\frac{2 \cdot 6}{2 {(12 x - 3)}^{\frac{1}{2}}}$

Step 3.13

Cancel the common factors.

Tap for more steps...

Step 3.13.1

Factor $2$ out of $2 {(12 x - 3)}^{\frac{1}{2}}$ .

$\frac{2 \cdot 6}{2 ({(12 x - 3)}^{\frac{1}{2}})}$

Step 3.13.2

Cancel the common factor.

$\frac{2 \cdot 6}{2 {(12 x - 3)}^{\frac{1}{2}}}$

Step 3.13.3

Rewrite the expression.

$\frac{6}{{(12 x - 3)}^{\frac{1}{2}}}$

Step 3.14

Evaluate the derivative at $x = 1$ .

$\frac{6}{{(12 (1) - 3)}^{\frac{1}{2}}}$

Step 3.15

Simplify.

Tap for more steps...

Step 3.15.1

Simplify the denominator.

Tap for more steps...

Step 3.15.1.1

Multiply $12$ by $1$ .

$\frac{6}{{(12 - 3)}^{\frac{1}{2}}}$

Step 3.15.1.2

Subtract $3$ from $12$ .

$\frac{6}{9^{\frac{1}{2}}}$

Step 3.15.1.3

Rewrite $9$ as $3^{2}$ .

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

Step 3.15.1.4

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.15.1.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.15.1.5.1

Cancel the common factor.

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

Step 3.15.1.5.2

Rewrite the expression.

$\frac{6}{3^{1}}$

Step 3.15.1.6

Evaluate the exponent.

$\frac{6}{3}$

Step 3.15.2

Divide $6$ by $3$ .

$2$

Step 4

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

Tap for more steps...

Step 4.1

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

$y - (3) = 2 \cdot (x - (1))$

Step 4.2

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

$y - 3 = 2 \cdot (x - 1)$

Step 4.3

Solve for $y$ .

Tap for more steps...

Step 4.3.1

Simplify $2 \cdot (x - 1)$ .

Tap for more steps...

Step 4.3.1.1

Rewrite.

$y - 3 = 0 + 0 + 2 \cdot (x - 1)$

Step 4.3.1.2

Simplify by adding zeros.

$y - 3 = 2 \cdot (x - 1)$

Step 4.3.1.3

Apply the distributive property.

$y - 3 = 2 x + 2 \cdot - 1$

Step 4.3.1.4

Multiply $2$ by $- 1$ .

$y - 3 = 2 x - 2$

Step 4.3.2

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

Tap for more steps...

Step 4.3.2.1

Add $3$ to both sides of the equation.

$y = 2 x - 2 + 3$

Step 4.3.2.2

Add $- 2$ and $3$ .

$y = 2 x + 1$

Step 5