Calculus Examples

y = \frac{8 x}{x^{2} + 1}

$y = \frac{8 x}{x^{2} + 1}$ at the origin and at the point

(1, 4)

$(1, 4)$

Step 1

Find the first derivative and evaluate at

x = 1

$x = 1$ and

y = 4

$y = 4$ to find the slope of the tangent line.

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Step 1.1

Since

8

$8$ is constant with respect to

x

$x$ , the derivative of

\frac{8 x}{x^{2} + 1}

$\frac{8 x}{x^{2} + 1}$ with respect to

x

$x$ is

8 \frac{d}{d x} [\frac{x}{x^{2} + 1}]

$8 \frac{d}{d x} [\frac{x}{x^{2} + 1}]$ .

8 \frac{d}{d x} [\frac{x}{x^{2} + 1}]

$8 \frac{d}{d x} [\frac{x}{x^{2} + 1}]$

Step 1.2

Differentiate using the Quotient Rule which states that

\frac{d}{d x} [\frac{f (x)}{g (x)}]

$\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is

\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}

$\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where

f (x) = x

$f (x) = x$ and

g (x) = x^{2} + 1

$g (x) = x^{2} + 1$ .

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

$8 \frac{(x^{2} + 1) \frac{d}{d x} [x] - x \frac{d}{d x} [x^{2} + 1]}{{(x^{2} + 1)}^{2}}$

Step 1.3

Differentiate.

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Step 1.3.1

Differentiate using the Power Rule which states that

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

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

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

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

$8 \frac{(x^{2} + 1) \cdot 1 - x \frac{d}{d x} [x^{2} + 1]}{{(x^{2} + 1)}^{2}}$

Step 1.3.2

Multiply

x^{2} + 1

$x^{2} + 1$ by

1

$1$ .

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

$8 \frac{x^{2} + 1 - x \frac{d}{d x} [x^{2} + 1]}{{(x^{2} + 1)}^{2}}$

Step 1.3.3

By the Sum Rule, the derivative of

x^{2} + 1

$x^{2} + 1$ with respect to

x

$x$ is

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

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

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

$8 \frac{x^{2} + 1 - x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1])}{{(x^{2} + 1)}^{2}}$

Step 1.3.4

Differentiate using the Power Rule which states that

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

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

n x^{n - 1}

$n x^{n - 1}$ where

n = 2

$n = 2$ .

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

$8 \frac{x^{2} + 1 - x (2 x + \frac{d}{d x} [1])}{{(x^{2} + 1)}^{2}}$

Step 1.3.5

Since

1

$1$ is constant with respect to

x

$x$ , the derivative of

1

$1$ with respect to

x

$x$ is

0

$0$ .

8 \frac{x^{2} + 1 - x (2 x + 0)}{{(x^{2} + 1)}^{2}}

$8 \frac{x^{2} + 1 - x (2 x + 0)}{{(x^{2} + 1)}^{2}}$

Step 1.3.6

Simplify the expression.

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Step 1.3.6.1

Add

2 x

$2 x$ and

0

$0$ .

8 \frac{x^{2} + 1 - x (2 x)}{{(x^{2} + 1)}^{2}}

$8 \frac{x^{2} + 1 - x (2 x)}{{(x^{2} + 1)}^{2}}$

Step 1.3.6.2

Multiply

2

$2$ by

- 1

$- 1$ .

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

$8 \frac{x^{2} + 1 - 2 x \cdot x}{{(x^{2} + 1)}^{2}}$

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

$8 \frac{x^{2} + 1 - 2 x \cdot x}{{(x^{2} + 1)}^{2}}$

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

$8 \frac{x^{2} + 1 - 2 x \cdot x}{{(x^{2} + 1)}^{2}}$

Step 1.4

Raise

x

$x$ to the power of

1

$1$ .

8 \frac{x^{2} + 1 - 2 (x^{1} x)}{{(x^{2} + 1)}^{2}}

$8 \frac{x^{2} + 1 - 2 (x^{1} x)}{{(x^{2} + 1)}^{2}}$

Step 1.5

Raise

x

$x$ to the power of

1

$1$ .

8 \frac{x^{2} + 1 - 2 (x^{1} x^{1})}{{(x^{2} + 1)}^{2}}

$8 \frac{x^{2} + 1 - 2 (x^{1} x^{1})}{{(x^{2} + 1)}^{2}}$

Step 1.6

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

8 \frac{x^{2} + 1 - 2 x^{1 + 1}}{{(x^{2} + 1)}^{2}}

$8 \frac{x^{2} + 1 - 2 x^{1 + 1}}{{(x^{2} + 1)}^{2}}$

Step 1.7

Add

1

$1$ and

1

$1$ .

8 \frac{x^{2} + 1 - 2 x^{2}}{{(x^{2} + 1)}^{2}}

$8 \frac{x^{2} + 1 - 2 x^{2}}{{(x^{2} + 1)}^{2}}$

Step 1.8

Subtract

2 x^{2}

$2 x^{2}$ from

x^{2}

$x^{2}$ .

8 \frac{- x^{2} + 1}{{(x^{2} + 1)}^{2}}

$8 \frac{- x^{2} + 1}{{(x^{2} + 1)}^{2}}$

Step 1.9

Combine

8

$8$ and

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

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

\frac{8 (- x^{2} + 1)}{{(x^{2} + 1)}^{2}}

$\frac{8 (- x^{2} + 1)}{{(x^{2} + 1)}^{2}}$

Step 1.10

Simplify.

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Step 1.10.1

Apply the distributive property.

\frac{8 (- x^{2}) + 8 \cdot 1}{{(x^{2} + 1)}^{2}}

$\frac{8 (- x^{2}) + 8 \cdot 1}{{(x^{2} + 1)}^{2}}$

Step 1.10.2

Simplify each term.

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Step 1.10.2.1

Multiply

- 1

$- 1$ by

8

$8$ .

\frac{- 8 x^{2} + 8 \cdot 1}{{(x^{2} + 1)}^{2}}

$\frac{- 8 x^{2} + 8 \cdot 1}{{(x^{2} + 1)}^{2}}$

Step 1.10.2.2

Multiply

8

$8$ by

1

$1$ .

\frac{- 8 x^{2} + 8}{{(x^{2} + 1)}^{2}}

$\frac{- 8 x^{2} + 8}{{(x^{2} + 1)}^{2}}$

\frac{- 8 x^{2} + 8}{{(x^{2} + 1)}^{2}}

$\frac{- 8 x^{2} + 8}{{(x^{2} + 1)}^{2}}$

\frac{- 8 x^{2} + 8}{{(x^{2} + 1)}^{2}}

$\frac{- 8 x^{2} + 8}{{(x^{2} + 1)}^{2}}$

Step 1.11

Evaluate the derivative at

x = 1

$x = 1$ .

\frac{- 8 {(1)}^{2} + 8}{{({(1)}^{2} + 1)}^{2}}

$\frac{- 8 {(1)}^{2} + 8}{{({(1)}^{2} + 1)}^{2}}$

Step 1.12

Simplify.

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Step 1.12.1

Simplify the numerator.

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Step 1.12.1.1

One to any power is one.

\frac{- 8 \cdot 1 + 8}{{(1^{2} + 1)}^{2}}

$\frac{- 8 \cdot 1 + 8}{{(1^{2} + 1)}^{2}}$

Step 1.12.1.2

Multiply

- 8

$- 8$ by

1

$1$ .

\frac{- 8 + 8}{{(1^{2} + 1)}^{2}}

$\frac{- 8 + 8}{{(1^{2} + 1)}^{2}}$

Step 1.12.1.3

Add

- 8

$- 8$ and

8

$8$ .

\frac{0}{{(1^{2} + 1)}^{2}}

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

\frac{0}{{(1^{2} + 1)}^{2}}

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

Step 1.12.2

Simplify the denominator.

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Step 1.12.2.1

One to any power is one.

\frac{0}{{(1 + 1)}^{2}}

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

Step 1.12.2.2

Add

1

$1$ and

1

$1$ .

\frac{0}{2^{2}}

$\frac{0}{2^{2}}$

Step 1.12.2.3

Raise

2

$2$ to the power of

2

$2$ .

\frac{0}{4}

$\frac{0}{4}$

\frac{0}{4}

$\frac{0}{4}$

Step 1.12.3

Divide

0

$0$ by

4

$4$ .

0

$0$

0

$0$

0

$0$

Step 2

Plug the slope and point values into the point-slope formula and solve for

y

$y$ .

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Step 2.1

Use the slope

0

$0$ and a given point

(1, 4)

$(1, 4)$ to substitute for

x_{1}

$x_{1}$ and

y_{1}

$y_{1}$ in the point-slope form

y - y_{1} = m (x - x_{1})

$y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation

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

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

y - (4) = 0 \cdot (x - (1))

$y - (4) = 0 \cdot (x - (1))$

Step 2.2

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

y - 4 = 0 \cdot (x - 1)

$y - 4 = 0 \cdot (x - 1)$

Step 2.3

Solve for

y

$y$ .

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Step 2.3.1

Multiply

0

$0$ by

x - 1

$x - 1$ .

y - 4 = 0

$y - 4 = 0$

Step 2.3.2

Add

4

$4$ to both sides of the equation.

y = 4

$y = 4$

y = 4

$y = 4$

y = 4

$y = 4$

Step 3