Calculus Examples

$y = \sqrt{3 + x^{3}}$ , $(1, 2)$

Step 1

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 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 + x^{3}}$ as ${(3 + x^{3})}^{\frac{1}{2}}$ .

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

Step 1.2

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 $g (x) = 3 + x^{3}$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $3 + x^{3}$ .

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

Step 1.2.2

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

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

Step 1.2.3

Replace all occurrences of $u$ with $3 + x^{3}$ .

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $2$ .

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

Step 1.6.2

Subtract $2$ from $1$ .

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

Step 1.7

Combine fractions.

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Step 1.7.1

Move the negative in front of the fraction.

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

Step 1.7.2

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

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

Step 1.7.3

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

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

Step 1.8

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

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

Step 1.9

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

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

Step 1.10

Add $0$ and $\frac{d}{d x} [x^{3}]$ .

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

Step 1.11

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 1.12

Combine fractions.

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Step 1.12.1

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

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

Step 1.12.2

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

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

Step 1.13

Evaluate the derivative at $x = 1$ .

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

Step 1.14

Simplify.

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Step 1.14.1

One to any power is one.

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

Step 1.14.2

Simplify the denominator.

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Step 1.14.2.1

One to any power is one.

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

Step 1.14.2.2

Add $3$ and $1$ .

$\frac{3 \cdot 1}{2 \cdot 4^{\frac{1}{2}}}$

Step 1.14.2.3

Rewrite $4$ as $2^{2}$ .

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

Step 1.14.2.4

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

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

Step 1.14.2.5

Cancel the common factor of $2$ .

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Step 1.14.2.5.1

Cancel the common factor.

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

Step 1.14.2.5.2

Rewrite the expression.

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

Step 1.14.2.6

Evaluate the exponent.

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

Step 1.14.3

Multiply.

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Step 1.14.3.1

Multiply $3$ by $1$ .

$\frac{3}{2 \cdot 2}$

Step 1.14.3.2

Multiply $2$ by $2$ .

$\frac{3}{4}$

Step 2

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

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Step 2.1

Use the slope $\frac{3}{4}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = \frac{3}{4} \cdot (x - (1))$

Step 2.2

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

$y - 2 = \frac{3}{4} \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{3}{4} \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y - 2 = 0 + 0 + \frac{3}{4} \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 2 = \frac{3}{4} \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y - 2 = \frac{3}{4} x + \frac{3}{4} \cdot - 1$

Step 2.3.1.4

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

$y - 2 = \frac{3 x}{4} + \frac{3}{4} \cdot - 1$

Step 2.3.1.5

Multiply $\frac{3}{4} \cdot - 1$ .

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Step 2.3.1.5.1

Combine $\frac{3}{4}$ and $- 1$ .

$y - 2 = \frac{3 x}{4} + \frac{3 \cdot - 1}{4}$

Step 2.3.1.5.2

Multiply $3$ by $- 1$ .

$y - 2 = \frac{3 x}{4} + \frac{- 3}{4}$

Step 2.3.1.6

Move the negative in front of the fraction.

$y - 2 = \frac{3 x}{4} - \frac{3}{4}$

Step 2.3.2

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

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Step 2.3.2.1

Add $2$ to both sides of the equation.

$y = \frac{3 x}{4} - \frac{3}{4} + 2$

Step 2.3.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{3 x}{4} - \frac{3}{4} + 2 \cdot \frac{4}{4}$

Step 2.3.2.3

Combine $2$ and $\frac{4}{4}$ .

$y = \frac{3 x}{4} - \frac{3}{4} + \frac{2 \cdot 4}{4}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{3 x}{4} + \frac{- 3 + 2 \cdot 4}{4}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $2$ by $4$ .

$y = \frac{3 x}{4} + \frac{- 3 + 8}{4}$

Step 2.3.2.5.2

Add $- 3$ and $8$ .

$y = \frac{3 x}{4} + \frac{5}{4}$

Step 2.3.3

Reorder terms.

$y = \frac{3}{4} x + \frac{5}{4}$

Step 3