Calculus Examples

$f (x) = \sqrt{x}$ , $x = \frac{1}{25}$

Step 1

Find the corresponding $y$ -value to $x = \frac{1}{25}$ .

Tap for more steps...

Step 1.1

Substitute $\frac{1}{25}$ in for $x$ .

$y = \sqrt{\frac{1}{25}}$

Step 1.2

Simplify $\sqrt{\frac{1}{25}}$ .

Tap for more steps...

Step 1.2.1

Remove parentheses.

$y = \sqrt{\frac{1}{25}}$

Step 1.2.2

Rewrite $\sqrt{\frac{1}{25}}$ as $\frac{\sqrt{1}}{\sqrt{25}}$ .

$y = \frac{\sqrt{1}}{\sqrt{25}}$

Step 1.2.3

Any root of $1$ is $1$ .

$y = \frac{1}{\sqrt{25}}$

Step 1.2.4

Simplify the denominator.

Tap for more steps...

Step 1.2.4.1

Rewrite $25$ as $5^{2}$ .

$y = \frac{1}{\sqrt{5^{2}}}$

Step 1.2.4.2

Pull terms out from under the radical, assuming positive real numbers.

$y = \frac{1}{5}$

Step 2

Find the first derivative and evaluate at $x = \frac{1}{25}$ and $y = \frac{1}{5}$ to find the slope of the tangent line.

Tap for more steps...

Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{d}{d x} [x^{\frac{1}{2}}]$

Step 2.2

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

$\frac{1}{2} x^{\frac{1}{2} - 1}$

Step 2.3

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

$\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}$

Step 2.4

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

$\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}$

Step 2.6

Simplify the numerator.

Tap for more steps...

Step 2.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} x^{\frac{1 - 2}{2}}$

Step 2.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} x^{\frac{- 1}{2}}$

Step 2.7

Move the negative in front of the fraction.

$\frac{1}{2} x^{- \frac{1}{2}}$

Step 2.8

Simplify.

Tap for more steps...

Step 2.8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}}$

Step 2.8.2

Multiply $\frac{1}{2}$ by $\frac{1}{x^{\frac{1}{2}}}$ .

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

Step 2.9

Evaluate the derivative at $x = \frac{1}{25}$ .

$\frac{1}{2 {(\frac{1}{25})}^{\frac{1}{2}}}$

Step 2.10

Simplify.

Tap for more steps...

Step 2.10.1

Simplify the denominator.

Tap for more steps...

Step 2.10.1.1

Apply the product rule to $\frac{1}{25}$ .

$\frac{1}{2 \frac{1^{\frac{1}{2}}}{25^{\frac{1}{2}}}}$

Step 2.10.1.2

One to any power is one.

$\frac{1}{2 \frac{1}{25^{\frac{1}{2}}}}$

Step 2.10.1.3

Simplify the denominator.

Tap for more steps...

Step 2.10.1.3.1

Rewrite $25$ as $5^{2}$ .

$\frac{1}{2 \frac{1}{{(5^{2})}^{\frac{1}{2}}}}$

Step 2.10.1.3.2

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

$\frac{1}{2 \frac{1}{5^{2 (\frac{1}{2})}}}$

Step 2.10.1.3.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.10.1.3.3.1

Cancel the common factor.

$\frac{1}{2 \frac{1}{5^{2 (\frac{1}{2})}}}$

Step 2.10.1.3.3.2

Rewrite the expression.

$\frac{1}{2 \frac{1}{5^{1}}}$

Step 2.10.1.3.4

Evaluate the exponent.

$\frac{1}{2 (\frac{1}{5})}$

Step 2.10.2

Combine $2$ and $\frac{1}{5}$ .

$\frac{1}{\frac{2}{5}}$

Step 2.10.3

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{5}{2})$

Step 2.10.4

Multiply $\frac{5}{2}$ by $1$ .

$\frac{5}{2}$

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 $\frac{5}{2}$ and a given point $(\frac{1}{25}, \frac{1}{5})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{1}{5}) = \frac{5}{2} \cdot (x - (\frac{1}{25}))$

Step 3.2

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

$y - \frac{1}{5} = \frac{5}{2} \cdot (x - \frac{1}{25})$

Step 3.3

Solve for $y$ .

Tap for more steps...

Step 3.3.1

Simplify $\frac{5}{2} \cdot (x - \frac{1}{25})$ .

Tap for more steps...

Step 3.3.1.1

Rewrite.

$y - \frac{1}{5} = 0 + 0 + \frac{5}{2} \cdot (x - \frac{1}{25})$

Step 3.3.1.2

Simplify by adding zeros.

$y - \frac{1}{5} = \frac{5}{2} \cdot (x - \frac{1}{25})$

Step 3.3.1.3

Apply the distributive property.

$y - \frac{1}{5} = \frac{5}{2} x + \frac{5}{2} (- \frac{1}{25})$

Step 3.3.1.4

Combine $\frac{5}{2}$ and $x$ .

$y - \frac{1}{5} = \frac{5 x}{2} + \frac{5}{2} (- \frac{1}{25})$

Step 3.3.1.5

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.3.1.5.1

Move the leading negative in $- \frac{1}{25}$ into the numerator.

$y - \frac{1}{5} = \frac{5 x}{2} + \frac{5}{2} \cdot \frac{- 1}{25}$

Step 3.3.1.5.2

Factor $5$ out of $25$ .

$y - \frac{1}{5} = \frac{5 x}{2} + \frac{5}{2} \cdot \frac{- 1}{5 (5)}$

Step 3.3.1.5.3

Cancel the common factor.

$y - \frac{1}{5} = \frac{5 x}{2} + \frac{5}{2} \cdot \frac{- 1}{5 \cdot 5}$

Step 3.3.1.5.4

Rewrite the expression.

$y - \frac{1}{5} = \frac{5 x}{2} + \frac{1}{2} \cdot \frac{- 1}{5}$

Step 3.3.1.6

Multiply $\frac{1}{2}$ by $\frac{- 1}{5}$ .

$y - \frac{1}{5} = \frac{5 x}{2} + \frac{- 1}{2 \cdot 5}$

Step 3.3.1.7

Simplify the expression.

Tap for more steps...

Step 3.3.1.7.1

Multiply $2$ by $5$ .

$y - \frac{1}{5} = \frac{5 x}{2} + \frac{- 1}{10}$

Step 3.3.1.7.2

Move the negative in front of the fraction.

$y - \frac{1}{5} = \frac{5 x}{2} - \frac{1}{10}$

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 $\frac{1}{5}$ to both sides of the equation.

$y = \frac{5 x}{2} - \frac{1}{10} + \frac{1}{5}$

Step 3.3.2.2

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{5 x}{2} - \frac{1}{10} + \frac{1}{5} \cdot \frac{2}{2}$

Step 3.3.2.3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.3.2.3.1

Multiply $\frac{1}{5}$ by $\frac{2}{2}$ .

$y = \frac{5 x}{2} - \frac{1}{10} + \frac{2}{5 \cdot 2}$

Step 3.3.2.3.2

Multiply $5$ by $2$ .

$y = \frac{5 x}{2} - \frac{1}{10} + \frac{2}{10}$

Step 3.3.2.4

Combine the numerators over the common denominator.

$y = \frac{5 x}{2} + \frac{- 1 + 2}{10}$

Step 3.3.2.5

Add $- 1$ and $2$ .

$y = \frac{5 x}{2} + \frac{1}{10}$

Step 3.3.3

Reorder terms.

$y = \frac{5}{2} x + \frac{1}{10}$

Step 4