Calculus Examples

$y = \frac{x^{2}}{5 + x}$ , $(1, \frac{1}{6})$

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = \frac{1}{6}$ to find the slope of the tangent line.

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Step 1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2}$ and $g (x) = 5 + x$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

Move $2$ to the left of $5 + x$ .

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

Step 1.2.3

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

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

Step 1.2.4

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

$\frac{2 (5 + x) x - x^{2} (0 + \frac{d}{d x} [x])}{{(5 + x)}^{2}}$

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.2.7

Multiply $- 1$ by $1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Simplify the numerator.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Multiply $2$ by $5$ .

$\frac{10 x + 2 x \cdot x - x^{2}}{{(5 + x)}^{2}}$

Step 1.3.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.3.1.2.1

Move $x$ .

$\frac{10 x + 2 (x \cdot x) - x^{2}}{{(5 + x)}^{2}}$

Step 1.3.3.1.2.2

Multiply $x$ by $x$ .

$\frac{10 x + 2 x^{2} - x^{2}}{{(5 + x)}^{2}}$

Step 1.3.3.2

Subtract $x^{2}$ from $2 x^{2}$ .

$\frac{10 x + x^{2}}{{(5 + x)}^{2}}$

Step 1.3.4

Reorder terms.

$\frac{x^{2} + 10 x}{{(x + 5)}^{2}}$

Step 1.3.5

Factor $x$ out of $x^{2} + 10 x$ .

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Step 1.3.5.1

Factor $x$ out of $x^{2}$ .

$\frac{x \cdot x + 10 x}{{(x + 5)}^{2}}$

Step 1.3.5.2

Factor $x$ out of $10 x$ .

$\frac{x \cdot x + x \cdot 10}{{(x + 5)}^{2}}$

Step 1.3.5.3

Factor $x$ out of $x \cdot x + x \cdot 10$ .

$\frac{x (x + 10)}{{(x + 5)}^{2}}$

Step 1.4

Evaluate the derivative at $x = 1$ .

$\frac{(1) ((1) + 10)}{{((1) + 5)}^{2}}$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $1 + 10$ by $1$ .

$\frac{1 + 10}{{(1 + 5)}^{2}}$

Step 1.5.2

Simplify the denominator.

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Step 1.5.2.1

Add $1$ and $5$ .

$\frac{1 + 10}{6^{2}}$

Step 1.5.2.2

Raise $6$ to the power of $2$ .

$\frac{1 + 10}{36}$

Step 1.5.3

Add $1$ and $10$ .

$\frac{11}{36}$

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 $\frac{11}{36}$ and a given point $(1, \frac{1}{6})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{1}{6}) = \frac{11}{36} \cdot (x - (1))$

Step 2.2

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

$y - \frac{1}{6} = \frac{11}{36} \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{11}{36} \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y - \frac{1}{6} = 0 + 0 + \frac{11}{36} \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - \frac{1}{6} = \frac{11}{36} \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y - \frac{1}{6} = \frac{11}{36} x + \frac{11}{36} \cdot - 1$

Step 2.3.1.4

Combine $\frac{11}{36}$ and $x$ .

$y - \frac{1}{6} = \frac{11 x}{36} + \frac{11}{36} \cdot - 1$

Step 2.3.1.5

Multiply $\frac{11}{36} \cdot - 1$ .

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Step 2.3.1.5.1

Combine $\frac{11}{36}$ and $- 1$ .

$y - \frac{1}{6} = \frac{11 x}{36} + \frac{11 \cdot - 1}{36}$

Step 2.3.1.5.2

Multiply $11$ by $- 1$ .

$y - \frac{1}{6} = \frac{11 x}{36} + \frac{- 11}{36}$

Step 2.3.1.6

Move the negative in front of the fraction.

$y - \frac{1}{6} = \frac{11 x}{36} - \frac{11}{36}$

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 $\frac{1}{6}$ to both sides of the equation.

$y = \frac{11 x}{36} - \frac{11}{36} + \frac{1}{6}$

Step 2.3.2.2

To write $\frac{1}{6}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$y = \frac{11 x}{36} - \frac{11}{36} + \frac{1}{6} \cdot \frac{6}{6}$

Step 2.3.2.3

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.2.3.1

Multiply $\frac{1}{6}$ by $\frac{6}{6}$ .

$y = \frac{11 x}{36} - \frac{11}{36} + \frac{6}{6 \cdot 6}$

Step 2.3.2.3.2

Multiply $6$ by $6$ .

$y = \frac{11 x}{36} - \frac{11}{36} + \frac{6}{36}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{11 x}{36} + \frac{- 11 + 6}{36}$

Step 2.3.2.5

Add $- 11$ and $6$ .

$y = \frac{11 x}{36} + \frac{- 5}{36}$

Step 2.3.2.6

Move the negative in front of the fraction.

$y = \frac{11 x}{36} - \frac{5}{36}$

Step 2.3.3

Reorder terms.

$y = \frac{11}{36} x - \frac{5}{36}$

Step 3