Calculus Examples

y = \sqrt{x + 3}

$y = \sqrt{x + 3}$ ,

$x = 1$

Step 1

Find the corresponding

$y$ -value to

$x = 1$ .

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Step 1.1

Substitute

$1$ in for

$x$ .

$y = \sqrt{(1) + 3}$

Step 1.2

Simplify

$\sqrt{(1) + 3}$ .

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Step 1.2.1

Add

$1$ and

$3$ .

$y = \sqrt{4}$

Step 1.2.2

Rewrite

$4$ as

$2^{2}$ .

$y = \sqrt{2^{2}}$

Step 1.2.3

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

$y = 2$

Step 2

Find the first derivative and evaluate at

$x = 1$ and

$y = 2$ to find the slope of the tangent line.

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Step 2.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{x + 3}$ as

${(x + 3)}^{\frac{1}{2}}$ .

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

Step 2.2

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

$g (x) = x + 3$ .

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Step 2.2.1

To apply the Chain Rule, set

$u$ as

$x + 3$ .

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

Step 2.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d u} [u^{n}]$ is

$n u^{n - 1}$ where

$n = \frac{1}{2}$ .

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

Step 2.2.3

Replace all occurrences of

$u$ with

$x + 3$ .

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

Step 2.3

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Step 2.4

Combine

$- 1$ and

$\frac{2}{2}$ .

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply

$- 1$ by

$2$ .

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

Step 2.6.2

Subtract

$2$ from

$1$ .

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

Step 2.7

Combine fractions.

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Step 2.7.1

Move the negative in front of the fraction.

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

Step 2.7.2

Combine

$\frac{1}{2}$ and

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

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

Step 2.7.3

Move

${(x + 3)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.8

By the Sum Rule, the derivative of

$x + 3$ with respect to

$x$ is

$\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

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

Step 2.9

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 2.10

Since

$3$ is constant with respect to

$x$ , the derivative of

$3$ with respect to

$x$ is

$0$ .

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

Step 2.11

Simplify the expression.

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Step 2.11.1

Add

$1$ and

$0$ .

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

Step 2.11.2

Multiply

$\frac{1}{2 {(x + 3)}^{\frac{1}{2}}}$ by

$1$ .

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

Step 2.12

Evaluate the derivative at

$x = 1$ .

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

Step 2.13

Simplify.

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Step 2.13.1

Simplify the denominator.

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Step 2.13.1.1

Add

$1$ and

$3$ .

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

Step 2.13.1.2

Rewrite

$4$ as

$2^{2}$ .

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

Step 2.13.1.3

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.13.1.4

Cancel the common factor of

$2$ .

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Step 2.13.1.4.1

Cancel the common factor.

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

Step 2.13.1.4.2

Rewrite the expression.

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

Step 2.13.1.5

Evaluate the exponent.

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

Step 2.13.2

Multiply

$2$ by

$2$ .

$\frac{1}{4}$

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

$\frac{1}{4}$ and a given point

$(1, 2)$ 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

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 3.2

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

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

Step 3.3

Solve for

$y$ .

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Step 3.3.1

Simplify

$\frac{1}{4} \cdot (x - 1)$ .

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Step 3.3.1.1

Rewrite.

$y - 2 = 0 + 0 + \frac{1}{4} \cdot (x - 1)$

Step 3.3.1.2

Simplify by adding zeros.

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

Step 3.3.1.3

Apply the distributive property.

$y - 2 = \frac{1}{4} x + \frac{1}{4} \cdot - 1$

Step 3.3.1.4

Combine

$\frac{1}{4}$ and

$x$ .

$y - 2 = \frac{x}{4} + \frac{1}{4} \cdot - 1$

Step 3.3.1.5

Combine

$\frac{1}{4}$ and

$- 1$ .

$y - 2 = \frac{x}{4} + \frac{- 1}{4}$

Step 3.3.1.6

Move the negative in front of the fraction.

$y - 2 = \frac{x}{4} - \frac{1}{4}$

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

$2$ to both sides of the equation.

$y = \frac{x}{4} - \frac{1}{4} + 2$

Step 3.3.2.2

To write

$2$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

$y = \frac{x}{4} - \frac{1}{4} + 2 \cdot \frac{4}{4}$

Step 3.3.2.3

Combine

$2$ and

$\frac{4}{4}$ .

$y = \frac{x}{4} - \frac{1}{4} + \frac{2 \cdot 4}{4}$

Step 3.3.2.4

Combine the numerators over the common denominator.

$y = \frac{x}{4} + \frac{- 1 + 2 \cdot 4}{4}$

Step 3.3.2.5

Simplify the numerator.

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Step 3.3.2.5.1

Multiply

$2$ by

$4$ .

$y = \frac{x}{4} + \frac{- 1 + 8}{4}$

Step 3.3.2.5.2

Add

$- 1$ and

$8$ .

$y = \frac{x}{4} + \frac{7}{4}$

Step 3.3.3

Reorder terms.

$y = \frac{1}{4} x + \frac{7}{4}$

Step 4