Calculus Examples

$y = \frac{2 x}{x^{2} + 1}$ ,

$(- 1, - 1)$

Step 1

Find the first derivative and evaluate at

$x = - 1$ and

$y = - 1$ to find the slope of the tangent line.

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Step 1.1

Since

$2$ is constant with respect to

$x$ , the derivative of

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

$x$ is

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

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

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

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

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

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

$n = 1$ .

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

Step 1.3.2

Multiply

$x^{2} + 1$ by

$1$ .

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

Step 1.3.3

By the Sum Rule, the derivative of

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

$x$ is

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

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

Step 1.3.4

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 2$ .

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

Step 1.3.5

Since

$1$ is constant with respect to

$x$ , the derivative of

$1$ with respect to

$x$ is

$0$ .

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

Step 1.3.6

Simplify the expression.

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Step 1.3.6.1

Add

$2 x$ and

$0$ .

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

Step 1.3.6.2

Multiply

$2$ by

$- 1$ .

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

Step 1.4

Raise

$x$ to the power of

$1$ .

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

Step 1.5

Raise

$x$ to the power of

$1$ .

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

Step 1.6

Use the power rule

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

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

Step 1.7

Add

$1$ and

$1$ .

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

Step 1.8

Subtract

$2 x^{2}$ from

$x^{2}$ .

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

Step 1.9

Combine

$2$ and

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

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

Step 1.10

Simplify.

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Step 1.10.1

Apply the distributive property.

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

Step 1.10.2

Simplify each term.

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Step 1.10.2.1

Multiply

$- 1$ by

$2$ .

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

Step 1.10.2.2

Multiply

$2$ by

$1$ .

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

Step 1.11

Evaluate the derivative at

$x = - 1$ .

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

Step 1.12

Simplify.

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Step 1.12.1

Simplify the numerator.

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Step 1.12.1.1

Raise

$- 1$ to the power of

$2$ .

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

Step 1.12.1.2

Multiply

$- 2$ by

$1$ .

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

Step 1.12.1.3

Add

$- 2$ and

$2$ .

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

Step 1.12.2

Simplify the denominator.

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Step 1.12.2.1

Raise

$- 1$ to the power of

$2$ .

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

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

$(- 1, - 1)$ 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

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

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

Step 2.2

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

$y + 1 = 0 \cdot (x + 1)$

Step 2.3

Solve for

$y$ .

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Step 2.3.1

Multiply

$0$ by

$x + 1$ .

$y + 1 = 0$

Step 2.3.2

Subtract

$1$ from both sides of the equation.

$y = - 1$

Step 3