#The Quotient Rule 🚀

Hey there, future calculus master! Ready to tackle the Quotient Rule? This is a key tool for differentiating those tricky rational functions, and you'll see it pop up all over the AP exam. Let's make sure you're totally comfortable with it!

# ⛳ Quotient Rule Definition

The Quotient Rule helps us find the derivative of a function that's a ratio of two other functions. Here's the formal definition:

$$\frac{d}{dx} \Bigg[\frac{f(x)}{g(x)}\Bigg] = \frac{g(x)\frac{d}{dx}f(x)-f(x)\frac{d}{dx}g(x)}{(g(x))^2}$$

Or, if you prefer a more compact version using u and v:

$$\frac{d}{dx} \Bigg[\frac{\textcolor{red}{f(x)}}{\textcolor{green}{g(x)}}\Bigg] = \frac{d}{dx} (\frac{\textcolor{red}{u}}{\textcolor{green}{v}}) = \frac{\textcolor{green}{v}\textcolor{purple}{u'}-\textcolor{red}{u}\textcolor{blue}{v'}}{\textcolor{green}{v^2}}$$

Key Point: The Quotient Rule only applies when you have a function in the form of a fraction, $\frac{f(x)}{g(x)}$, and both f(x) and g(x) are differentiable. ✅

#✏️ Quotient Rule Walkthrough

Let's see it in action!

Example: Find the derivative of $y=\frac{x^2+x-2}{x^3+6}$.

1. Identify f(x) and g(x):

   • $f(x) = x^2 + x - 2$
   • $g(x) = x^3 + 6$

2. Find the derivatives:

   • $f'(x) = 2x + 1$
   • $g'(x) = 3x^2$

3. Apply the Quotient Rule:

   $$\frac{dy}{dx} = \frac{(x^3+6)(2x+1)-(x^2+x-2)(3x^2)}{(x^3+6)^2}$$

4. Simplify (this is where your algebra skills come in!):

   $$\frac{dy}{dx} = \frac{(2x^4+x^3+12x+6)-(3x^4+3x^3-6x^2)}{(x^3+6)^2}$$

   $$\frac{dy}{dx} = \frac{-x^4-2x^3+6x^2+12x+6}{(x^3+6)^2}$$

#🧮 Quotient Rule: Practice Problems

Time to test your skills! Try these examples before looking at the solutions.

#Quotient Rule: Example 1

Find the derivative of $y = \frac{e^x}{1+x^2}$.

Solution:

1. $f(x) = e^x$, so $f'(x) = e^x$

2. $g(x) = 1 + x^2$, so $g'(x) = 2x$

3. Apply the Quotient Rule:

   $$\frac{dy}{dx} = \frac{(1+x^2)(e^x) - (e^x)(2x)}{(1+x^2)^2}$$

4. Simplify:

   $$\frac{dy}{dx} = \frac{e^x + x^2e^x - 2xe^x}{(1+x^2)^2} = \frac{e^x(1 - 2x + x^2)}{(1+x^2)^2} = \frac{e^x(1-x)^2}{(1+x^2)^2}$$

#Quotient Rule: Example 2

Find the derivative of $y = \frac{\sin(x)}{1+\cos(x)}$.

Solution:

1. $f(x) = \sin(x)$, so $f'(x) = \cos(x)$

2. $g(x) = 1 + \cos(x)$, so $g'(x) = -\sin(x)$

3. Apply the Quotient Rule:

   $$\frac{dy}{dx} = \frac{(1+\cos(x))(\cos(x)) - (\sin(x))(-\sin(x))}{(1+\cos(x))^2}$$

4. Simplify:

   $$\frac{dy}{dx} = \frac{\cos(x) + \cos^2(x) + \sin^2(x)}{(1+\cos(x))^2}$$

<math-block>\frac{dy}{dx} = \frac{\cos(x) + 1}{(1+\cos(x))^2} = \frac{1}{1+\cos(x)}</math-block>

#💫 Closing

You've got this! The Quotient Rule might seem tricky at first, but with practice, it'll become second nature. Remember to take your time, double-check your work, and don't be afraid to break down complex problems into smaller steps. 🎉

Here's a quick recap:

Image Courtesy of OnLine Math Learning

Practice Question