Calculus Examples

$y = (\frac{10 x^{5} - 16 x^{2}}{x^{3}})$

Step 1

Remove parentheses.

$y = \frac{10 x^{5} - 16 x^{2}}{x^{3}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{10 x^{5} - 16 x^{2}}{x^{3}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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 $f (x) = 10 x^{5} - 16 x^{2}$ and $g (x) = x^{3}$ .

$\frac{x^{3} \frac{d}{d x} [10 x^{5} - 16 x^{2}] - (10 x^{5} - 16 x^{2}) \frac{d}{d x} [x^{3}]}{{(x^{3})}^{2}}$

Step 4.2

Differentiate.

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Step 4.2.1

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

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Step 4.2.1.1

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

$\frac{x^{3} \frac{d}{d x} [10 x^{5} - 16 x^{2}] - (10 x^{5} - 16 x^{2}) \frac{d}{d x} [x^{3}]}{x^{3 \cdot 2}}$

Step 4.2.1.2

Multiply $3$ by $2$ .

$\frac{x^{3} \frac{d}{d x} [10 x^{5} - 16 x^{2}] - (10 x^{5} - 16 x^{2}) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 4.2.2

By the Sum Rule, the derivative of $10 x^{5} - 16 x^{2}$ with respect to $x$ is $\frac{d}{d x} [10 x^{5}] + \frac{d}{d x} [- 16 x^{2}]$ .

$\frac{x^{3} (\frac{d}{d x} [10 x^{5}] + \frac{d}{d x} [- 16 x^{2}]) - (10 x^{5} - 16 x^{2}) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 4.2.3

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{5}$ with respect to $x$ is $10 \frac{d}{d x} [x^{5}]$ .

$\frac{x^{3} (10 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 16 x^{2}]) - (10 x^{5} - 16 x^{2}) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 4.2.4

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

$\frac{x^{3} (10 (5 x^{4}) + \frac{d}{d x} [- 16 x^{2}]) - (10 x^{5} - 16 x^{2}) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 4.2.5

Multiply $5$ by $10$ .

$\frac{x^{3} (50 x^{4} + \frac{d}{d x} [- 16 x^{2}]) - (10 x^{5} - 16 x^{2}) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 4.2.6

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16 x^{2}$ with respect to $x$ is $- 16 \frac{d}{d x} [x^{2}]$ .

$\frac{x^{3} (50 x^{4} - 16 \frac{d}{d x} [x^{2}]) - (10 x^{5} - 16 x^{2}) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 4.2.7

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

$\frac{x^{3} (50 x^{4} - 16 (2 x)) - (10 x^{5} - 16 x^{2}) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 4.2.8

Multiply $2$ by $- 16$ .

$\frac{x^{3} (50 x^{4} - 32 x) - (10 x^{5} - 16 x^{2}) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 4.2.9

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

$\frac{x^{3} (50 x^{4} - 32 x) - (10 x^{5} - 16 x^{2}) (3 x^{2})}{x^{6}}$

Step 4.2.10

Simplify with factoring out.

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Step 4.2.10.1

Multiply $3$ by $- 1$ .

$\frac{x^{3} (50 x^{4} - 32 x) - 3 (10 x^{5} - 16 x^{2}) x^{2}}{x^{6}}$

Step 4.2.10.2

Factor $x^{2}$ out of $x^{3} (50 x^{4} - 32 x) - 3 (10 x^{5} - 16 x^{2}) x^{2}$ .

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Step 4.2.10.2.1

Factor $x^{2}$ out of $x^{3} (50 x^{4} - 32 x)$ .

$\frac{x^{2} (x (50 x^{4} - 32 x)) - 3 (10 x^{5} - 16 x^{2}) x^{2}}{x^{6}}$

Step 4.2.10.2.2

Factor $x^{2}$ out of $- 3 (10 x^{5} - 16 x^{2}) x^{2}$ .

$\frac{x^{2} (x (50 x^{4} - 32 x)) + x^{2} (- 3 (10 x^{3} - 16) x^{2})}{x^{6}}$

Step 4.2.10.2.3

Factor $x^{2}$ out of $x^{2} (x (50 x^{4} - 32 x)) + x^{2} (- 3 (10 x^{3} - 16) x^{2})$ .

$\frac{x^{2} (x (50 x^{4} - 32 x) - 3 (10 x^{3} - 16) x^{2})}{x^{6}}$

Step 4.3

Cancel the common factors.

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Step 4.3.1

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

$\frac{x^{2} (x (50 x^{4} - 32 x) - 3 (10 x^{3} - 16) x^{2})}{x^{2} x^{4}}$

Step 4.3.2

Cancel the common factor.

$\frac{x^{2} (x (50 x^{4} - 32 x) - 3 (10 x^{3} - 16) x^{2})}{x^{2} x^{4}}$

Step 4.3.3

Rewrite the expression.

$\frac{x (50 x^{4} - 32 x) - 3 (10 x^{3} - 16) x^{2}}{x^{4}}$

Step 4.4

Factor $x$ out of $x (50 x^{4} - 32 x) - 3 (10 x^{3} - 16) x^{2}$ .

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Step 4.4.1

Factor $x$ out of $- 3 (10 x^{3} - 16) x^{2}$ .

$\frac{x (50 x^{4} - 32 x) + x (- 3 (10 x^{3} - 16) x)}{x^{4}}$

Step 4.4.2

Factor $x$ out of $x (50 x^{4} - 32 x) + x (- 3 (10 x^{3} - 16) x)$ .

$\frac{x (50 x^{4} - 32 x - 3 (10 x^{3} - 16) x)}{x^{4}}$

Step 4.5

Cancel the common factors.

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Step 4.5.1

Factor $x$ out of $x^{4}$ .

$\frac{x (50 x^{4} - 32 x - 3 (10 x^{3} - 16) x)}{x \cdot x^{3}}$

Step 4.5.2

Cancel the common factor.

$\frac{x (50 x^{4} - 32 x - 3 (10 x^{3} - 16) x)}{x \cdot x^{3}}$

Step 4.5.3

Rewrite the expression.

$\frac{50 x^{4} - 32 x - 3 (10 x^{3} - 16) x}{x^{3}}$

Step 4.6

Factor $x$ out of $50 x^{4} - 32 x - 3 (10 x^{3} - 16) x$ .

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Step 4.6.1

Factor $x$ out of $50 x^{4}$ .

$\frac{x (50 x^{3}) - 32 x - 3 (10 x^{3} - 16) x}{x^{3}}$

Step 4.6.2

Factor $x$ out of $- 32 x$ .

$\frac{x (50 x^{3}) + x \cdot - 32 - 3 (10 x^{3} - 16) x}{x^{3}}$

Step 4.6.3

Factor $x$ out of $- 3 (10 x^{3} - 16) x$ .

$\frac{x (50 x^{3}) + x \cdot - 32 + x (- 3 (10 x^{3} - 16))}{x^{3}}$

Step 4.6.4

Factor $x$ out of $x (50 x^{3}) + x \cdot - 32$ .

$\frac{x (50 x^{3} - 32) + x (- 3 (10 x^{3} - 16))}{x^{3}}$

Step 4.6.5

Factor $x$ out of $x (50 x^{3} - 32) + x (- 3 (10 x^{3} - 16))$ .

$\frac{x (50 x^{3} - 32 - 3 (10 x^{3} - 16))}{x^{3}}$

Step 4.7

Cancel the common factors.

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Step 4.7.1

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

$\frac{x (50 x^{3} - 32 - 3 (10 x^{3} - 16))}{x \cdot x^{2}}$

Step 4.7.2

Cancel the common factor.

$\frac{x (50 x^{3} - 32 - 3 (10 x^{3} - 16))}{x \cdot x^{2}}$

Step 4.7.3

Rewrite the expression.

$\frac{50 x^{3} - 32 - 3 (10 x^{3} - 16)}{x^{2}}$

Step 4.8

Simplify.

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Step 4.8.1

Apply the distributive property.

$\frac{50 x^{3} - 32 - 3 (10 x^{3}) - 3 \cdot - 16}{x^{2}}$

Step 4.8.2

Simplify the numerator.

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Step 4.8.2.1

Simplify each term.

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Step 4.8.2.1.1

Multiply $10$ by $- 3$ .

$\frac{50 x^{3} - 32 - 30 x^{3} - 3 \cdot - 16}{x^{2}}$

Step 4.8.2.1.2

Multiply $- 3$ by $- 16$ .

$\frac{50 x^{3} - 32 - 30 x^{3} + 48}{x^{2}}$

Step 4.8.2.2

Subtract $30 x^{3}$ from $50 x^{3}$ .

$\frac{20 x^{3} - 32 + 48}{x^{2}}$

Step 4.8.2.3

Add $- 32$ and $48$ .

$\frac{20 x^{3} + 16}{x^{2}}$

Step 4.8.3

Factor $4$ out of $20 x^{3} + 16$ .

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Step 4.8.3.1

Factor $4$ out of $20 x^{3}$ .

$\frac{4 (5 x^{3}) + 16}{x^{2}}$

Step 4.8.3.2

Factor $4$ out of $16$ .

$\frac{4 (5 x^{3}) + 4 (4)}{x^{2}}$

Step 4.8.3.3

Factor $4$ out of $4 (5 x^{3}) + 4 (4)$ .

$\frac{4 (5 x^{3} + 4)}{x^{2}}$

Step 5

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

$y' = \frac{4 (5 x^{3} + 4)}{x^{2}}$

Step 6

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

$\frac{d y}{d x} = \frac{4 (5 x^{3} + 4)}{x^{2}}$