Calculus Examples

$y = \frac{8 x}{x^{2} + 1}$ at the origin and at the point $(1, 4)$

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = 4$ to find the slope of the tangent line.

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Step 1.1

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8 x}{x^{2} + 1}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{x}{x^{2} + 1}]$ .

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

Step 1.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = x^{2} + 1$ .

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

Step 1.3

Differentiate.

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Step 1.3.1

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

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

Step 1.3.2

Multiply $x^{2} + 1$ by $1$ .

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

Step 1.3.3

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

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

Step 1.3.4

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

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

Step 1.3.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 1.3.6

Simplify the expression.

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Step 1.3.6.1

Add $2 x$ and $0$ .

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

Step 1.3.6.2

Multiply $2$ by $- 1$ .

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

Step 1.4

Raise $x$ to the power of $1$ .

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

Step 1.5

Raise $x$ to the power of $1$ .

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

Step 1.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.7

Add $1$ and $1$ .

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

Step 1.8

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

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

Step 1.9

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

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

Step 1.10

Simplify.

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Step 1.10.1

Apply the distributive property.

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

Step 1.10.2

Simplify each term.

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Step 1.10.2.1

Multiply $- 1$ by $8$ .

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

Step 1.10.2.2

Multiply $8$ by $1$ .

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

Step 1.11

Evaluate the derivative at $x = 1$ .

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

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.

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

Step 1.12.1.2

Multiply $- 8$ by $1$ .

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

Step 1.12.1.3

Add $- 8$ and $8$ .

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

Step 1.12.2

Simplify the denominator.

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Step 1.12.2.1

One to any power is one.

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

Step 1.12.2.2

Add $1$ and $1$ .

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

Step 1.12.2.3

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

$\frac{0}{4}$

Step 1.12.3

Divide $0$ by $4$ .

$0$

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 $0$ and a given point $(1, 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

Multiply $0$ by $x - 1$ .

$y - 4 = 0$

Step 2.3.2

Add $4$ to both sides of the equation.

$y = 4$

Step 3