Calculus Examples

$y = \frac{x}{8 x + 3}$

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 $f (x) = x$ and $g (x) = 8 x + 3$ .

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

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 $n = 1$ .

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

Step 2.2

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

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

Step 3

Evaluate $\frac{d}{d x} [8 x]$ .

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Step 3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

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

Step 3.2

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

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

Step 3.3

Multiply $8$ by $1$ .

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

Step 4

Differentiate using the Constant Rule.

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Step 4.1

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

$\frac{(8 x + 3) \cdot 1 - x (8 + 0)}{{(8 x + 3)}^{2}}$

Step 4.2

Add $8$ and $0$ .

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Simplify the numerator.

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Step 5.2.1

Combine the opposite terms in $8 x \cdot 1 + 3 \cdot 1 - x \cdot 8$ .

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Step 5.2.1.1

Reorder the factors in the terms $8 x \cdot 1$ and $- x \cdot 8$ .

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

Step 5.2.1.2

Subtract $8 x$ from $1 \cdot 8 x$ .

$\frac{3 \cdot 1 + 0}{{(8 x + 3)}^{2}}$

Step 5.2.1.3

Add $3 \cdot 1$ and $0$ .

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

Step 5.2.2

Multiply $3$ by $1$ .

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