Calculus Examples

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

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = 6 x^{5}$ and $g (y) = 5 y^{4}$ .

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

Step 2

Differentiate.

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Step 2.1

Since $6 x^{5}$ is constant with respect to $y$ , the derivative of $6 x^{5}$ with respect to $y$ is $0$ .

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

Step 2.2

Since $5$ is constant with respect to $y$ , the derivative of $5 y^{4}$ with respect to $y$ is $5 \frac{d}{d y} [y^{4}]$ .

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

Step 2.3

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

$\frac{5 y^{4} \cdot 0 - 1 \cdot 6 x^{5} (5 (4 y^{3}))}{{(5 y^{4})}^{2}}$

Step 2.4

Multiply $4$ by $5$ .

$\frac{5 y^{4} \cdot 0 - 1 \cdot 6 x^{5} (20 y^{3})}{{(5 y^{4})}^{2}}$

Step 3

Simplify.

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Step 3.1

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

$\frac{5 y^{4} \cdot 0 - 1 \cdot 6 x^{5} (20 y^{3})}{5^{2} {(y^{4})}^{2}}$

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

Multiply $5 y^{4} \cdot 0$ .

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Step 3.2.1.1.1

Multiply $0$ by $5$ .

$\frac{0 y^{4} - 1 \cdot 6 x^{5} (20 y^{3})}{5^{2} {(y^{4})}^{2}}$

Step 3.2.1.1.2

Multiply $0$ by $y^{4}$ .

$\frac{0 - 1 \cdot 6 x^{5} (20 y^{3})}{5^{2} {(y^{4})}^{2}}$

Step 3.2.1.2

Rewrite using the commutative property of multiplication.

$\frac{0 - 1 \cdot 6 \cdot 20 x^{5} y^{3}}{5^{2} {(y^{4})}^{2}}$

Step 3.2.1.3

Multiply $- 1 \cdot 6 \cdot 20$ .

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Step 3.2.1.3.1

Multiply $- 1$ by $6$ .

$\frac{0 - 6 \cdot 20 x^{5} y^{3}}{5^{2} {(y^{4})}^{2}}$

Step 3.2.1.3.2

Multiply $- 6$ by $20$ .

$\frac{0 - 120 x^{5} y^{3}}{5^{2} {(y^{4})}^{2}}$

Step 3.2.2

Subtract $120 x^{5} y^{3}$ from $0$ .

$\frac{- 120 x^{5} y^{3}}{5^{2} {(y^{4})}^{2}}$

Step 3.3

Combine terms.

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Step 3.3.1

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

$\frac{- 120 x^{5} y^{3}}{25 {(y^{4})}^{2}}$

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, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{- 120 x^{5} y^{3}}{25 y^{4 \cdot 2}}$

Step 3.3.2.2

Multiply $4$ by $2$ .

$\frac{- 120 x^{5} y^{3}}{25 y^{8}}$

Step 3.3.3

Cancel the common factor of $- 120$ and $25$ .

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Step 3.3.3.1

Factor $5$ out of $- 120 x^{5} y^{3}$ .

$\frac{5 (- 24 x^{5} y^{3})}{25 y^{8}}$

Step 3.3.3.2

Cancel the common factors.

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Step 3.3.3.2.1

Factor $5$ out of $25 y^{8}$ .

$\frac{5 (- 24 x^{5} y^{3})}{5 (5 y^{8})}$

Step 3.3.3.2.2

Cancel the common factor.

$\frac{5 (- 24 x^{5} y^{3})}{5 (5 y^{8})}$

Step 3.3.3.2.3

Rewrite the expression.

$\frac{- 24 x^{5} y^{3}}{5 y^{8}}$

Step 3.3.4

Cancel the common factor of $y^{3}$ and $y^{8}$ .

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Step 3.3.4.1

Factor $y^{3}$ out of $- 24 x^{5} y^{3}$ .

$\frac{y^{3} (- 24 x^{5})}{5 y^{8}}$

Step 3.3.4.2

Cancel the common factors.

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Step 3.3.4.2.1

Factor $y^{3}$ out of $5 y^{8}$ .

$\frac{y^{3} (- 24 x^{5})}{y^{3} (5 y^{5})}$

Step 3.3.4.2.2

Cancel the common factor.

$\frac{y^{3} (- 24 x^{5})}{y^{3} (5 y^{5})}$

Step 3.3.4.2.3

Rewrite the expression.

$\frac{- 24 x^{5}}{5 y^{5}}$

Step 3.3.5

Move the negative in front of the fraction.

$- \frac{24 x^{5}}{5 y^{5}}$