Calculus Examples

$y = \frac{x}{3 - 2 x^{2}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{x}{3 - 2 x^{2}})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $g (x) = 3 - 2 x^{2}$ .

$\frac{(3 - 2 x^{2}) \frac{d}{d x} [x] - x \frac{d}{d x} [3 - 2 x^{2}]}{{(3 - 2 x^{2})}^{2}}$

Step 3.2

Differentiate.

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Step 3.2.1

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

$\frac{(3 - 2 x^{2}) \cdot 1 - x \frac{d}{d x} [3 - 2 x^{2}]}{{(3 - 2 x^{2})}^{2}}$

Step 3.2.2

Multiply $3 - 2 x^{2}$ by $1$ .

$\frac{3 - 2 x^{2} - x \frac{d}{d x} [3 - 2 x^{2}]}{{(3 - 2 x^{2})}^{2}}$

Step 3.2.3

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

$\frac{3 - 2 x^{2} - x (\frac{d}{d x} [3] + \frac{d}{d x} [- 2 x^{2}])}{{(3 - 2 x^{2})}^{2}}$

Step 3.2.4

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

$\frac{3 - 2 x^{2} - x (0 + \frac{d}{d x} [- 2 x^{2}])}{{(3 - 2 x^{2})}^{2}}$

Step 3.2.5

Add $0$ and $\frac{d}{d x} [- 2 x^{2}]$ .

$\frac{3 - 2 x^{2} - x \frac{d}{d x} [- 2 x^{2}]}{{(3 - 2 x^{2})}^{2}}$

Step 3.2.6

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

$\frac{3 - 2 x^{2} - x (- 2 \frac{d}{d x} [x^{2}])}{{(3 - 2 x^{2})}^{2}}$

Step 3.2.7

Multiply $- 2$ by $- 1$ .

$\frac{3 - 2 x^{2} + 2 x \frac{d}{d x} [x^{2}]}{{(3 - 2 x^{2})}^{2}}$

Step 3.2.8

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

$\frac{3 - 2 x^{2} + 2 x (2 x)}{{(3 - 2 x^{2})}^{2}}$

Step 3.2.9

Multiply $2$ by $2$ .

$\frac{3 - 2 x^{2} + 4 x \cdot x}{{(3 - 2 x^{2})}^{2}}$

Step 3.3

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

$\frac{3 - 2 x^{2} + 4 (x^{1} x)}{{(3 - 2 x^{2})}^{2}}$

Step 3.4

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

$\frac{3 - 2 x^{2} + 4 (x^{1} x^{1})}{{(3 - 2 x^{2})}^{2}}$

Step 3.5

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

$\frac{3 - 2 x^{2} + 4 x^{1 + 1}}{{(3 - 2 x^{2})}^{2}}$

Step 3.6

Add $1$ and $1$ .

$\frac{3 - 2 x^{2} + 4 x^{2}}{{(3 - 2 x^{2})}^{2}}$

Step 3.7

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

$\frac{3 + 2 x^{2}}{{(3 - 2 x^{2})}^{2}}$

Step 3.8

Reorder terms.

$\frac{2 x^{2} + 3}{{(- 2 x^{2} + 3)}^{2}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{2 x^{2} + 3}{{(- 2 x^{2} + 3)}^{2}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{2 x^{2} + 3}{{(- 2 x^{2} + 3)}^{2}}$