Calculus Examples

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

Step 1

This derivative could not be completed using the chain rule. Mathway will use another method.

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d a} [\frac{f (a)}{g (a)}]$ is $\frac{g (a) \frac{d}{d a} [f (a)] - f (a) \frac{d}{d a} [g (a)]}{{g (a)}^{2}}$ where $f (a) = a x - 2$ and $g (a) = x^{3} - a$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $x$ by $1$ .

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

Step 3.5

Since $- 2$ is constant with respect to $a$ , the derivative of $- 2$ with respect to $a$ is $0$ .

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

Step 3.6

Add $x$ and $0$ .

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

Step 3.7

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

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

Step 3.8

Since $x^{3}$ is constant with respect to $a$ , the derivative of $x^{3}$ with respect to $a$ is $0$ .

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

Step 3.9

Add $0$ and $\frac{d}{d a} [- a]$ .

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

Step 3.10

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

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

Step 3.11

Multiply.

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Step 3.11.1

Multiply $- 1$ by $- 1$ .

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

Step 3.11.2

Multiply $a x - 2$ by $1$ .

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

Step 3.12

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

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

Step 3.13

Multiply $a x - 2$ by $1$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Simplify the numerator.

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Step 4.2.1

Combine the opposite terms in $x^{3} x - a x + a x - 2$ .

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Step 4.2.1.1

Add $- a x$ and $a x$ .

$\frac{x^{3} x + 0 - 2}{{(x^{3} - a)}^{2}}$

Step 4.2.1.2

Add $x^{3} x$ and $0$ .

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

Step 4.2.2

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 4.2.2.1

Multiply $x^{3}$ by $x$ .

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Step 4.2.2.1.1

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

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

Step 4.2.2.1.2

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

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

Step 4.2.2.2

Add $3$ and $1$ .

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