Calculus Examples

$\frac{6 x^{5}}{5 y^{4}}$

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 $f (x) = 6 x^{5}$ and $g (x) = 5 y^{4}$ .

$\frac{5 y^{4} \frac{d}{d x} [6 x^{5}] - 1 \cdot 6 x^{5} \frac{d}{d x} [5 y^{4}]}{{(5 y^{4})}^{2}}$

Step 2

Differentiate.

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Step 2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{5}$ with respect to $x$ is $6 \frac{d}{d x} [x^{5}]$ .

$\frac{5 y^{4} (6 \frac{d}{d x} [x^{5}]) - 1 \cdot 6 x^{5} \frac{d}{d x} [5 y^{4}]}{{(5 y^{4})}^{2}}$

Step 2.2

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

$\frac{5 y^{4} (6 (5 x^{4})) - 1 \cdot 6 x^{5} \frac{d}{d x} [5 y^{4}]}{{(5 y^{4})}^{2}}$

Step 2.3

Multiply $5$ by $6$ .

$\frac{5 y^{4} (30 x^{4}) - 1 \cdot 6 x^{5} \frac{d}{d x} [5 y^{4}]}{{(5 y^{4})}^{2}}$

Step 2.4

Since $5 y^{4}$ is constant with respect to $x$ , the derivative of $5 y^{4}$ with respect to $x$ is $0$ .

$\frac{5 y^{4} (30 x^{4}) - 1 \cdot 6 x^{5} \cdot 0}{{(5 y^{4})}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the product rule to $5 y^{4}$ .

$\frac{5 y^{4} (30 x^{4}) - 1 \cdot 6 x^{5} \cdot 0}{5^{2} {(y^{4})}^{2}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{5 \cdot 30 y^{4} x^{4} - 1 \cdot 6 x^{5} \cdot 0}{5^{2} {(y^{4})}^{2}}$

Step 3.2.1.2

Multiply $5$ by $30$ .

$\frac{150 y^{4} x^{4} - 1 \cdot 6 x^{5} \cdot 0}{5^{2} {(y^{4})}^{2}}$

Step 3.2.1.3

Multiply $- 1$ by $6$ .

$\frac{150 y^{4} x^{4} - 6 x^{5} \cdot 0}{5^{2} {(y^{4})}^{2}}$

Step 3.2.1.4

Multiply $- 6 x^{5} \cdot 0$ .

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Step 3.2.1.4.1

Multiply $0$ by $- 6$ .

$\frac{150 y^{4} x^{4} + 0 x^{5}}{5^{2} {(y^{4})}^{2}}$

Step 3.2.1.4.2

Multiply $0$ by $x^{5}$ .

$\frac{150 y^{4} x^{4} + 0}{5^{2} {(y^{4})}^{2}}$

Step 3.2.2

Add $150 y^{4} x^{4}$ and $0$ .

$\frac{150 y^{4} x^{4}}{5^{2} {(y^{4})}^{2}}$

Step 3.3

Combine terms.

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Step 3.3.1

Raise $5$ to the power of $2$ .

$\frac{150 y^{4} x^{4}}{25 {(y^{4})}^{2}}$

Step 3.3.2

Multiply the exponents in ${(y^{4})}^{2}$ .

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Step 3.3.2.1

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

$\frac{150 y^{4} x^{4}}{25 y^{4 \cdot 2}}$

Step 3.3.2.2

Multiply $4$ by $2$ .

$\frac{150 y^{4} x^{4}}{25 y^{8}}$

Step 3.3.3

Cancel the common factor of $150$ and $25$ .

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Step 3.3.3.1

Factor $25$ out of $150 y^{4} x^{4}$ .

$\frac{25 (6 y^{4} x^{4})}{25 y^{8}}$

Step 3.3.3.2

Cancel the common factors.

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Step 3.3.3.2.1

Factor $25$ out of $25 y^{8}$ .

$\frac{25 (6 y^{4} x^{4})}{25 (y^{8})}$

Step 3.3.3.2.2

Cancel the common factor.

$\frac{25 (6 y^{4} x^{4})}{25 y^{8}}$

Step 3.3.3.2.3

Rewrite the expression.

$\frac{6 y^{4} x^{4}}{y^{8}}$

Step 3.3.4

Cancel the common factor of $y^{4}$ and $y^{8}$ .

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Step 3.3.4.1

Factor $y^{4}$ out of $6 y^{4} x^{4}$ .

$\frac{y^{4} (6 x^{4})}{y^{8}}$

Step 3.3.4.2

Cancel the common factors.

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Step 3.3.4.2.1

Factor $y^{4}$ out of $y^{8}$ .

$\frac{y^{4} (6 x^{4})}{y^{4} y^{4}}$

Step 3.3.4.2.2

Cancel the common factor.

$\frac{y^{4} (6 x^{4})}{y^{4} y^{4}}$

Step 3.3.4.2.3

Rewrite the expression.

$\frac{6 x^{4}}{y^{4}}$