Calculus Examples

$y = \frac{3 x - 4}{x^{3}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{3 x - 4}{x^{3}})$

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 $f (x) = 3 x - 4$ and $g (x) = x^{3}$ .

$\frac{x^{3} \frac{d}{d x} [3 x - 4] - (3 x - 4) \frac{d}{d x} [x^{3}]}{{(x^{3})}^{2}}$

Step 3.2

Differentiate.

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Step 3.2.1

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 3.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{3} \frac{d}{d x} [3 x - 4] - (3 x - 4) \frac{d}{d x} [x^{3}]}{x^{3 \cdot 2}}$

Step 3.2.1.2

Multiply $3$ by $2$ .

$\frac{x^{3} \frac{d}{d x} [3 x - 4] - (3 x - 4) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3.2.2

By the Sum Rule, the derivative of $3 x - 4$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 4]$ .

$\frac{x^{3} (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 4]) - (3 x - 4) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3.2.3

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

$\frac{x^{3} (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 4]) - (3 x - 4) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3.2.4

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

$\frac{x^{3} (3 \cdot 1 + \frac{d}{d x} [- 4]) - (3 x - 4) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3.2.5

Multiply $3$ by $1$ .

$\frac{x^{3} (3 + \frac{d}{d x} [- 4]) - (3 x - 4) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3.2.6

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$\frac{x^{3} (3 + 0) - (3 x - 4) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3.2.7

Simplify the expression.

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Step 3.2.7.1

Add $3$ and $0$ .

$\frac{x^{3} \cdot 3 - (3 x - 4) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3.2.7.2

Move $3$ to the left of $x^{3}$ .

$\frac{3 \cdot x^{3} - (3 x - 4) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3.2.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{3 x^{3} - (3 x - 4) (3 x^{2})}{x^{6}}$

Step 3.2.9

Multiply $3$ by $- 1$ .

$\frac{3 x^{3} - 3 (3 x - 4) x^{2}}{x^{6}}$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$\frac{3 x^{3} + (- 3 (3 x) - 3 \cdot - 4) x^{2}}{x^{6}}$

Step 3.3.2

Apply the distributive property.

$\frac{3 x^{3} - 3 (3 x) x^{2} - 3 \cdot - 4 x^{2}}{x^{6}}$

Step 3.3.3

Simplify the numerator.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.3.3.1.1.1

Move $x^{2}$ .

$\frac{3 x^{3} - 3 (3 (x^{2} x)) - 3 \cdot - 4 x^{2}}{x^{6}}$

Step 3.3.3.1.1.2

Multiply $x^{2}$ by $x$ .

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Step 3.3.3.1.1.2.1

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

$\frac{3 x^{3} - 3 (3 (x^{2} x^{1})) - 3 \cdot - 4 x^{2}}{x^{6}}$

Step 3.3.3.1.1.2.2

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

$\frac{3 x^{3} - 3 (3 x^{2 + 1}) - 3 \cdot - 4 x^{2}}{x^{6}}$

Step 3.3.3.1.1.3

Add $2$ and $1$ .

$\frac{3 x^{3} - 3 (3 x^{3}) - 3 \cdot - 4 x^{2}}{x^{6}}$

Step 3.3.3.1.2

Multiply $3$ by $- 3$ .

$\frac{3 x^{3} - 9 x^{3} - 3 \cdot - 4 x^{2}}{x^{6}}$

Step 3.3.3.1.3

Multiply $- 3$ by $- 4$ .

$\frac{3 x^{3} - 9 x^{3} + 12 x^{2}}{x^{6}}$

Step 3.3.3.2

Subtract $9 x^{3}$ from $3 x^{3}$ .

$\frac{- 6 x^{3} + 12 x^{2}}{x^{6}}$

Step 3.3.4

Factor $6 x^{2}$ out of $- 6 x^{3} + 12 x^{2}$ .

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Step 3.3.4.1

Factor $6 x^{2}$ out of $- 6 x^{3}$ .

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

Step 3.3.4.2

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

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

Step 3.3.4.3

Factor $6 x^{2}$ out of $6 x^{2} (- x) + 6 x^{2} (2)$ .

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

Step 3.3.5

Cancel the common factor of $x^{2}$ and $x^{6}$ .

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Step 3.3.5.1

Factor $x^{2}$ out of $6 x^{2} (- x + 2)$ .

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

Step 3.3.5.2

Cancel the common factors.

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Step 3.3.5.2.1

Factor $x^{2}$ out of $x^{6}$ .

$\frac{x^{2} (6 (- x + 2))}{x^{2} x^{4}}$

Step 3.3.5.2.2

Cancel the common factor.

$\frac{x^{2} (6 (- x + 2))}{x^{2} x^{4}}$

Step 3.3.5.2.3

Rewrite the expression.

$\frac{6 (- x + 2)}{x^{4}}$

Step 3.3.6

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

$\frac{6 (- (x) + 2)}{x^{4}}$

Step 3.3.7

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 3.3.8

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

$\frac{6 (- (x - 2))}{x^{4}}$

Step 3.3.9

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

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

Step 3.3.10

Move the negative in front of the fraction.

$- \frac{6 (x - 2)}{x^{4}}$

Step 4

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

$y' = - \frac{6 (x - 2)}{x^{4}}$

Step 5

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

$\frac{d y}{d x} = - \frac{6 (x - 2)}{x^{4}}$