Calculus Examples

$y = \frac{3 x^{5} - 7 x^{4} + 6}{x^{3}}$

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 $f (x) = 3 x^{5} - 7 x^{4} + 6$ and $g (x) = x^{3}$ .

$\frac{x^{3} \frac{d}{d x} [3 x^{5} - 7 x^{4} + 6] - (3 x^{5} - 7 x^{4} + 6) \frac{d}{d x} [x^{3}]}{{(x^{3})}^{2}}$

Step 2

Differentiate.

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Step 2.1

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

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Step 2.1.1

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

$\frac{x^{3} \frac{d}{d x} [3 x^{5} - 7 x^{4} + 6] - (3 x^{5} - 7 x^{4} + 6) \frac{d}{d x} [x^{3}]}{x^{3 \cdot 2}}$

Step 2.1.2

Multiply $3$ by $2$ .

$\frac{x^{3} \frac{d}{d x} [3 x^{5} - 7 x^{4} + 6] - (3 x^{5} - 7 x^{4} + 6) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.2

By the Sum Rule, the derivative of $3 x^{5} - 7 x^{4} + 6$ with respect to $x$ is $\frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [- 7 x^{4}] + \frac{d}{d x} [6]$ .

$\frac{x^{3} (\frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [- 7 x^{4}] + \frac{d}{d x} [6]) - (3 x^{5} - 7 x^{4} + 6) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{5}$ with respect to $x$ is $3 \frac{d}{d x} [x^{5}]$ .

$\frac{x^{3} (3 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 7 x^{4}] + \frac{d}{d x} [6]) - (3 x^{5} - 7 x^{4} + 6) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{x^{3} (3 (5 x^{4}) + \frac{d}{d x} [- 7 x^{4}] + \frac{d}{d x} [6]) - (3 x^{5} - 7 x^{4} + 6) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.5

Multiply $5$ by $3$ .

$\frac{x^{3} (15 x^{4} + \frac{d}{d x} [- 7 x^{4}] + \frac{d}{d x} [6]) - (3 x^{5} - 7 x^{4} + 6) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.6

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x^{4}$ with respect to $x$ is $- 7 \frac{d}{d x} [x^{4}]$ .

$\frac{x^{3} (15 x^{4} - 7 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [6]) - (3 x^{5} - 7 x^{4} + 6) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{x^{3} (15 x^{4} - 7 (4 x^{3}) + \frac{d}{d x} [6]) - (3 x^{5} - 7 x^{4} + 6) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.8

Multiply $4$ by $- 7$ .

$\frac{x^{3} (15 x^{4} - 28 x^{3} + \frac{d}{d x} [6]) - (3 x^{5} - 7 x^{4} + 6) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.9

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$\frac{x^{3} (15 x^{4} - 28 x^{3} + 0) - (3 x^{5} - 7 x^{4} + 6) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.10

Add $15 x^{4} - 28 x^{3}$ and $0$ .

$\frac{x^{3} (15 x^{4} - 28 x^{3}) - (3 x^{5} - 7 x^{4} + 6) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.11

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

$\frac{x^{3} (15 x^{4} - 28 x^{3}) - (3 x^{5} - 7 x^{4} + 6) (3 x^{2})}{x^{6}}$

Step 2.12

Simplify with factoring out.

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Step 2.12.1

Multiply $3$ by $- 1$ .

$\frac{x^{3} (15 x^{4} - 28 x^{3}) - 3 (3 x^{5} - 7 x^{4} + 6) x^{2}}{x^{6}}$

Step 2.12.2

Factor $x^{2}$ out of $x^{3} (15 x^{4} - 28 x^{3}) - 3 (3 x^{5} - 7 x^{4} + 6) x^{2}$ .

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Step 2.12.2.1

Factor $x^{2}$ out of $x^{3} (15 x^{4} - 28 x^{3})$ .

$\frac{x^{2} (x (15 x^{4} - 28 x^{3})) - 3 (3 x^{5} - 7 x^{4} + 6) x^{2}}{x^{6}}$

Step 2.12.2.2

Factor $x^{2}$ out of $- 3 (3 x^{5} - 7 x^{4} + 6) x^{2}$ .

$\frac{x^{2} (x (15 x^{4} - 28 x^{3})) + x^{2} (- 3 (3 x^{5} - 7 x^{4} + 6))}{x^{6}}$

Step 2.12.2.3

Factor $x^{2}$ out of $x^{2} (x (15 x^{4} - 28 x^{3})) + x^{2} (- 3 (3 x^{5} - 7 x^{4} + 6))$ .

$\frac{x^{2} (x (15 x^{4} - 28 x^{3}) - 3 (3 x^{5} - 7 x^{4} + 6))}{x^{6}}$

Step 3

Cancel the common factors.

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Step 3.1

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

$\frac{x^{2} (x (15 x^{4} - 28 x^{3}) - 3 (3 x^{5} - 7 x^{4} + 6))}{x^{2} x^{4}}$

Step 3.2

Cancel the common factor.

$\frac{x^{2} (x (15 x^{4} - 28 x^{3}) - 3 (3 x^{5} - 7 x^{4} + 6))}{x^{2} x^{4}}$

Step 3.3

Rewrite the expression.

$\frac{x (15 x^{4} - 28 x^{3}) - 3 (3 x^{5} - 7 x^{4} + 6)}{x^{4}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{x (15 x^{4}) + x (- 28 x^{3}) - 3 (3 x^{5} - 7 x^{4} + 6)}{x^{4}}$

Step 4.2

Apply the distributive property.

$\frac{x (15 x^{4}) + x (- 28 x^{3}) - 3 (3 x^{5}) - 3 (- 7 x^{4}) - 3 \cdot 6}{x^{4}}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{15 x \cdot x^{4} + x (- 28 x^{3}) - 3 (3 x^{5}) - 3 (- 7 x^{4}) - 3 \cdot 6}{x^{4}}$

Step 4.3.1.2

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 4.3.1.2.1

Move $x^{4}$ .

$\frac{15 (x^{4} x) + x (- 28 x^{3}) - 3 (3 x^{5}) - 3 (- 7 x^{4}) - 3 \cdot 6}{x^{4}}$

Step 4.3.1.2.2

Multiply $x^{4}$ by $x$ .

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Step 4.3.1.2.2.1

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

$\frac{15 (x^{4} x^{1}) + x (- 28 x^{3}) - 3 (3 x^{5}) - 3 (- 7 x^{4}) - 3 \cdot 6}{x^{4}}$

Step 4.3.1.2.2.2

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

$\frac{15 x^{4 + 1} + x (- 28 x^{3}) - 3 (3 x^{5}) - 3 (- 7 x^{4}) - 3 \cdot 6}{x^{4}}$

Step 4.3.1.2.3

Add $4$ and $1$ .

$\frac{15 x^{5} + x (- 28 x^{3}) - 3 (3 x^{5}) - 3 (- 7 x^{4}) - 3 \cdot 6}{x^{4}}$

Step 4.3.1.3

Rewrite using the commutative property of multiplication.

$\frac{15 x^{5} - 28 x \cdot x^{3} - 3 (3 x^{5}) - 3 (- 7 x^{4}) - 3 \cdot 6}{x^{4}}$

Step 4.3.1.4

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

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Step 4.3.1.4.1

Move $x^{3}$ .

$\frac{15 x^{5} - 28 (x^{3} x) - 3 (3 x^{5}) - 3 (- 7 x^{4}) - 3 \cdot 6}{x^{4}}$

Step 4.3.1.4.2

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

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Step 4.3.1.4.2.1

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

$\frac{15 x^{5} - 28 (x^{3} x^{1}) - 3 (3 x^{5}) - 3 (- 7 x^{4}) - 3 \cdot 6}{x^{4}}$

Step 4.3.1.4.2.2

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

$\frac{15 x^{5} - 28 x^{3 + 1} - 3 (3 x^{5}) - 3 (- 7 x^{4}) - 3 \cdot 6}{x^{4}}$

Step 4.3.1.4.3

Add $3$ and $1$ .

$\frac{15 x^{5} - 28 x^{4} - 3 (3 x^{5}) - 3 (- 7 x^{4}) - 3 \cdot 6}{x^{4}}$

Step 4.3.1.5

Multiply $3$ by $- 3$ .

$\frac{15 x^{5} - 28 x^{4} - 9 x^{5} - 3 (- 7 x^{4}) - 3 \cdot 6}{x^{4}}$

Step 4.3.1.6

Multiply $- 7$ by $- 3$ .

$\frac{15 x^{5} - 28 x^{4} - 9 x^{5} + 21 x^{4} - 3 \cdot 6}{x^{4}}$

Step 4.3.1.7

Multiply $- 3$ by $6$ .

$\frac{15 x^{5} - 28 x^{4} - 9 x^{5} + 21 x^{4} - 18}{x^{4}}$

Step 4.3.2

Subtract $9 x^{5}$ from $15 x^{5}$ .

$\frac{6 x^{5} - 28 x^{4} + 21 x^{4} - 18}{x^{4}}$

Step 4.3.3

Add $- 28 x^{4}$ and $21 x^{4}$ .

$\frac{6 x^{5} - 7 x^{4} - 18}{x^{4}}$