Calculus Examples

$\frac{x^{2}}{8 - 3 x}$

Step 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 $f (x) = x^{2}$ and $g (x) = 8 - 3 x$ .

$\frac{(8 - 3 x) \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [8 - 3 x]}{{(8 - 3 x)}^{2}}$

Step 2

Differentiate.

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Step 2.1

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

$\frac{(8 - 3 x) (2 x) - x^{2} \frac{d}{d x} [8 - 3 x]}{{(8 - 3 x)}^{2}}$

Step 2.2

Move $2$ to the left of $8 - 3 x$ .

$\frac{2 \cdot (8 - 3 x) x - x^{2} \frac{d}{d x} [8 - 3 x]}{{(8 - 3 x)}^{2}}$

Step 2.3

By the Sum Rule, the derivative of $8 - 3 x$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [- 3 x]$ .

$\frac{2 (8 - 3 x) x - x^{2} (\frac{d}{d x} [8] + \frac{d}{d x} [- 3 x])}{{(8 - 3 x)}^{2}}$

Step 2.4

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$\frac{2 (8 - 3 x) x - x^{2} (0 + \frac{d}{d x} [- 3 x])}{{(8 - 3 x)}^{2}}$

Step 2.5

Add $0$ and $\frac{d}{d x} [- 3 x]$ .

$\frac{2 (8 - 3 x) x - x^{2} \frac{d}{d x} [- 3 x]}{{(8 - 3 x)}^{2}}$

Step 2.6

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$\frac{2 (8 - 3 x) x - x^{2} (- 3 \frac{d}{d x} [x])}{{(8 - 3 x)}^{2}}$

Step 2.7

Multiply $- 3$ by $- 1$ .

$\frac{2 (8 - 3 x) x + 3 x^{2} \frac{d}{d x} [x]}{{(8 - 3 x)}^{2}}$

Step 2.8

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

$\frac{2 (8 - 3 x) x + 3 x^{2} \cdot 1}{{(8 - 3 x)}^{2}}$

Step 2.9

Multiply $3$ by $1$ .

$\frac{2 (8 - 3 x) x + 3 x^{2}}{{(8 - 3 x)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(2 \cdot 8 + 2 (- 3 x)) x + 3 x^{2}}{{(8 - 3 x)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{2 \cdot 8 x + 2 (- 3 x) x + 3 x^{2}}{{(8 - 3 x)}^{2}}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $2$ by $8$ .

$\frac{16 x + 2 (- 3 x) x + 3 x^{2}}{{(8 - 3 x)}^{2}}$

Step 3.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.1.2.1

Move $x$ .

$\frac{16 x + 2 (- 3 (x \cdot x)) + 3 x^{2}}{{(8 - 3 x)}^{2}}$

Step 3.3.1.2.2

Multiply $x$ by $x$ .

$\frac{16 x + 2 (- 3 x^{2}) + 3 x^{2}}{{(8 - 3 x)}^{2}}$

Step 3.3.1.3

Multiply $- 3$ by $2$ .

$\frac{16 x - 6 x^{2} + 3 x^{2}}{{(8 - 3 x)}^{2}}$

Step 3.3.2

Add $- 6 x^{2}$ and $3 x^{2}$ .

$\frac{16 x - 3 x^{2}}{{(8 - 3 x)}^{2}}$

Step 3.4

Reorder terms.

$\frac{- 3 x^{2} + 16 x}{{(- 3 x + 8)}^{2}}$

Step 3.5

Factor $x$ out of $- 3 x^{2} + 16 x$ .

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Step 3.5.1

Factor $x$ out of $- 3 x^{2}$ .

$\frac{x (- 3 x) + 16 x}{{(- 3 x + 8)}^{2}}$

Step 3.5.2

Factor $x$ out of $16 x$ .

$\frac{x (- 3 x) + x \cdot 16}{{(- 3 x + 8)}^{2}}$

Step 3.5.3

Factor $x$ out of $x (- 3 x) + x \cdot 16$ .

$\frac{x (- 3 x + 16)}{{(- 3 x + 8)}^{2}}$

Step 3.6

Factor $- 1$ out of $- 3 x$ .

$\frac{x (- (3 x) + 16)}{{(- 3 x + 8)}^{2}}$

Step 3.7

Rewrite $16$ as $- 1 (- 16)$ .

$\frac{x (- (3 x) - 1 (- 16))}{{(- 3 x + 8)}^{2}}$

Step 3.8

Factor $- 1$ out of $- (3 x) - 1 (- 16)$ .

$\frac{x (- (3 x - 16))}{{(- 3 x + 8)}^{2}}$

Step 3.9

Rewrite $- (3 x - 16)$ as $- 1 (3 x - 16)$ .

$\frac{x (- 1 (3 x - 16))}{{(- 3 x + 8)}^{2}}$

Step 3.10

Move the negative in front of the fraction.

$- \frac{x (3 x - 16)}{{(- 3 x + 8)}^{2}}$