Calculus Examples

$f (x) = 3 - \frac{6 y}{1 - 2 y^{2}}$

Step 1

Differentiate.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $3 - \frac{6 y}{1 - 2 y^{2}}$ with respect to $y$ is $\frac{d}{d y} [3] + \frac{d}{d y} [- \frac{6 y}{1 - 2 y^{2}}]$ .

$\frac{d}{d y} [3] + \frac{d}{d y} [- \frac{6 y}{1 - 2 y^{2}}]$

Step 1.2

Since $3$ is constant with respect to $y$ , the derivative of $3$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [- \frac{6 y}{1 - 2 y^{2}}]$

Step 2

Evaluate $\frac{d}{d y} [- \frac{6 y}{1 - 2 y^{2}}]$ .

Tap for more steps...

Step 2.1

Since $- 6$ is constant with respect to $y$ , the derivative of $- \frac{6 y}{1 - 2 y^{2}}$ with respect to $y$ is $- 6 \frac{d}{d y} [\frac{y}{1 - 2 y^{2}}]$ .

$0 - 6 \frac{d}{d y} [\frac{y}{1 - 2 y^{2}}]$

Step 2.2

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

$0 - 6 \frac{(1 - 2 y^{2}) \frac{d}{d y} [y] - y \frac{d}{d y} [1 - 2 y^{2}]}{{(1 - 2 y^{2})}^{2}}$

Step 2.3

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

$0 - 6 \frac{(1 - 2 y^{2}) \cdot 1 - y \frac{d}{d y} [1 - 2 y^{2}]}{{(1 - 2 y^{2})}^{2}}$

Step 2.4

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

$0 - 6 \frac{(1 - 2 y^{2}) \cdot 1 - y (\frac{d}{d y} [1] + \frac{d}{d y} [- 2 y^{2}])}{{(1 - 2 y^{2})}^{2}}$

Step 2.5

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

$0 - 6 \frac{(1 - 2 y^{2}) \cdot 1 - y (0 + \frac{d}{d y} [- 2 y^{2}])}{{(1 - 2 y^{2})}^{2}}$

Step 2.6

Since $- 2$ is constant with respect to $y$ , the derivative of $- 2 y^{2}$ with respect to $y$ is $- 2 \frac{d}{d y} [y^{2}]$ .

$0 - 6 \frac{(1 - 2 y^{2}) \cdot 1 - y (0 - 2 \frac{d}{d y} [y^{2}])}{{(1 - 2 y^{2})}^{2}}$

Step 2.7

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

$0 - 6 \frac{(1 - 2 y^{2}) \cdot 1 - y (0 - 2 (2 y))}{{(1 - 2 y^{2})}^{2}}$

Step 2.8

Multiply $1 - 2 y^{2}$ by $1$ .

$0 - 6 \frac{1 - 2 y^{2} - y (0 - 2 (2 y))}{{(1 - 2 y^{2})}^{2}}$

Step 2.9

Multiply $2$ by $- 2$ .

$0 - 6 \frac{1 - 2 y^{2} - y (0 - 4 y)}{{(1 - 2 y^{2})}^{2}}$

Step 2.10

Subtract $4 y$ from $0$ .

$0 - 6 \frac{1 - 2 y^{2} - y (- 4 y)}{{(1 - 2 y^{2})}^{2}}$

Step 2.11

Multiply $- 4$ by $- 1$ .

$0 - 6 \frac{1 - 2 y^{2} + 4 y \cdot y}{{(1 - 2 y^{2})}^{2}}$

Step 2.12

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

$0 - 6 \frac{1 - 2 y^{2} + 4 (y^{1} y)}{{(1 - 2 y^{2})}^{2}}$

Step 2.13

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

$0 - 6 \frac{1 - 2 y^{2} + 4 (y^{1} y^{1})}{{(1 - 2 y^{2})}^{2}}$

Step 2.14

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

$0 - 6 \frac{1 - 2 y^{2} + 4 y^{1 + 1}}{{(1 - 2 y^{2})}^{2}}$

Step 2.15

Add $1$ and $1$ .

$0 - 6 \frac{1 - 2 y^{2} + 4 y^{2}}{{(1 - 2 y^{2})}^{2}}$

Step 2.16

Add $- 2 y^{2}$ and $4 y^{2}$ .

$0 - 6 \frac{1 + 2 y^{2}}{{(1 - 2 y^{2})}^{2}}$

Step 2.17

Combine $- 6$ and $\frac{1 + 2 y^{2}}{{(1 - 2 y^{2})}^{2}}$ .

$0 + \frac{- 6 (1 + 2 y^{2})}{{(1 - 2 y^{2})}^{2}}$

Step 2.18

Move the negative in front of the fraction.

$0 - \frac{6 (1 + 2 y^{2})}{{(1 - 2 y^{2})}^{2}}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Apply the distributive property.

$0 - \frac{6 \cdot 1 + 6 (2 y^{2})}{{(1 - 2 y^{2})}^{2}}$

Step 3.2

Combine terms.

Tap for more steps...

Step 3.2.1

Multiply $6$ by $1$ .

$0 - \frac{6 + 6 (2 y^{2})}{{(1 - 2 y^{2})}^{2}}$

Step 3.2.2

Multiply $2$ by $6$ .

$0 - \frac{6 + 12 y^{2}}{{(1 - 2 y^{2})}^{2}}$

Step 3.2.3

Subtract $\frac{6 + 12 y^{2}}{{(1 - 2 y^{2})}^{2}}$ from $0$ .

$- \frac{6 + 12 y^{2}}{{(1 - 2 y^{2})}^{2}}$

Step 3.3

Factor $6$ out of $6 + 12 y^{2}$ .

Tap for more steps...

Step 3.3.1

Factor $6$ out of $6$ .

$- \frac{6 (1) + 12 y^{2}}{{(1 - 2 y^{2})}^{2}}$

Step 3.3.2

Factor $6$ out of $12 y^{2}$ .

$- \frac{6 (1) + 6 (2 y^{2})}{{(1 - 2 y^{2})}^{2}}$

Step 3.3.3

Factor $6$ out of $6 (1) + 6 (2 y^{2})$ .

$- \frac{6 (1 + 2 y^{2})}{{(1 - 2 y^{2})}^{2}}$