#Derivatives of Quotients

#Table of Contents

1. Introduction to Derivatives of Quotients
2. The Quotient Rule
3. Alternative Approach Using Product Rule
4. Exam Tips
5. Worked Examples
6. Practice Questions
7. Glossary
8. Summary and Key Takeaways

#Introduction to Derivatives of Quotients

Understanding how to find the derivative of one function divided by another is crucial in calculus. This process is governed by the quotient rule.

#The Quotient Rule

The Quotient Rule is a method for differentiating the quotient of two functions.

This can also be written in terms of $y$, $u$, and $v$: $$y = \frac{u}{v} \implies \frac{dy}{dx} = \frac{\frac{du}{dx} \cdot v - u \cdot \frac{dv}{dx}}{v^2}$$

#Alternative Approach Using Product Rule

Any quotient rule problem can alternatively be solved using the product rule by rewriting $f(x) = \frac{g(x)}{h(x)}$ as $f(x) = g(x) \cdot [h(x)]^{-1}$. The product rule and chain rule can then be applied.

#Exam Tips

Product Rule: If $y = u \cdot v$, then $$y' = u'v + uv'$$

#Worked Examples

#Example 1:

Find the derivative of $f(x) = \frac{4x + 3}{2x + 5}$.

Solution:

1. Assign $u$ and $v$ to each function: $u = 4x + 3$ $v = 2x + 5$

2. Find the derivatives: $u' = 4$ $v' = 2$

3. Apply the quotient rule: $$f'(x) = \frac{u'v - uv'}{v^2} = \frac{4(2x + 5) - (4x + 3) \cdot 2}{(2x + 5)^2}$$

4. Simplify: $$f'(x) = \frac{(8x + 20) - (8x + 6)}{(2x + 5)^2} = \frac{14}{(2x + 5)^2}$$

#Example 2:

Find the derivative of $g(x) = \frac{\sin 3x}{e^{4x}}$.

Solution:

1. Assign $u$ and $v$ to each function: $u = \sin 3x$ $v = e^{4x}$

2. Find the derivatives: $u' = 3 \cos 3x$ $v' = 4e^{4x}$

3. Apply the quotient rule: $$g'(x) = \frac{u'v - uv'}{v^2} = \frac{3 \cos 3x \cdot e^{4x} - \sin 3x \cdot 4e^{4x}}{(e^{4x})^2}$$

4. Simplify: $$g'(x) = \frac{e^{4x}(3 \cos 3x - 4 \sin 3x)}{(e^{4x})^2} = \frac{3 \cos 3x - 4 \sin 3x}{e^{4x}}$$

#Practice Questions

Practice Question

1. Differentiate $h(x) = \frac{x^2 + 1}{x - 2}$.
2. Differentiate $k(x) = \frac{\ln x}{x^2}$.
3. Differentiate $m(x) = \frac{e^x}{\cos x}$.

#Glossary

• Quotient Rule: A rule for differentiating the quotient of two functions.
• Product Rule: A rule for differentiating the product of two functions.
• Chain Rule: A rule for differentiating compositions of functions.

#Summary and Key Takeaways

• The quotient rule provides a method to differentiate the quotient of two functions.
• The numerator in the quotient rule can be obtained similarly to the product rule but with subtraction.
• The quotient rule can be replaced with the product rule by rewriting the quotient as a product.
• Practice applying these rules to become proficient in differentiating complex functions.

#Key Takeaways:

1. Understand and memorize the quotient rule formula.
2. Remember that the numerator is similar to the product rule but involves subtraction.
3. Practice differentiating using both the quotient rule and the alternative product rule approach.