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.

Tap for more steps...

Step 1.1

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 1.1.1

Move the negative in front of the fraction.

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

Step 1.1.2

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.

Tap for more steps...

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.

Tap for more steps...

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

Move the negative in front of the fraction.

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

Step 1.11

Simplify.

Tap for more steps...

Step 1.11.1

Apply the distributive property.

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

Step 1.11.2

Simplify each term.

Tap for more steps...

Step 1.11.2.1

Multiply $- 1$ by $8$ .

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

Step 1.11.2.2

Multiply $8$ by $1$ .

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

Step 1.12

Evaluate the derivative at $x = 1$ .

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

Step 1.13

Simplify.

Tap for more steps...

Step 1.13.1

Simplify the numerator.

Tap for more steps...

Step 1.13.1.1

One to any power is one.

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

Step 1.13.1.2

Multiply $- 8$ by $1$ .

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

Step 1.13.1.3

Add $- 8$ and $8$ .

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

Step 1.13.2

Simplify the denominator.

Tap for more steps...

Step 1.13.2.1

One to any power is one.

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

Step 1.13.2.2

Add $1$ and $1$ .

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

Step 1.13.2.3

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

$- \frac{0}{4}$

Step 1.13.3

Simplify the expression.

Tap for more steps...

Step 1.13.3.1

Divide $0$ by $4$ .

$- 0$

Step 1.13.3.2

Multiply $- 1$ by $0$ .

$0$

Step 2

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

Tap for more steps...

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 $y$ .

Tap for more steps...

Step 2.3.1

Multiply $0$ by $x - 1$ .

$y + 4 = 0$

Step 2.3.2

Subtract $4$ from both sides of the equation.

$y = - 4$

Step 3