Welcome back, students! As we approach the Spring 2025 Chapter 3 Test, it is time to solidify your understanding of derivatives. This review sheet covers the essential differentiation rules and applications you will face on the exam.

To help you study effectively, I have transcribed the practice problems below and grouped them by the techniques required to solve them. Remember, identifying which rule to use is half the battle!

1. Simplification Before Differentiation

Before jumping into a complex rule, look to see if algebra can make your life easier.

  • Problem 1: $$y = \frac{9x^2-5x-4}{\sqrt{x}}$$

Strategy: Instead of the Quotient Rule, rewrite the denominator as $x^{1/2}$ and split the fraction into three separate terms using power rules. Then, differentiation is simple subtraction of exponents.

2. The Big Three: Product, Quotient, and Chain Rules

You must be comfortable mixing and matching these rules. Pay close attention to trigonometric functions and logarithms.

  • Product Rule: Used in Problem 2: $$f(x) = (5x^2 - 3x + 2)(\sin(x))$$
  • Quotient Rule: Used in Problem 3: $$g(x) = \frac{3x^4+5}{\cos(x)}$$
  • Chain Rule & Log Properties: Look at Problem 8: $$y = \log_3 \frac{(4x-2)^6}{3x}$$

Pro Tip for #8: Use the properties of logarithms to expand this expression before deriving. It turns a nasty Chain/Quotient rule problem into a simple sum and difference problem!

3. Transcendental Functions

Make sure you have memorized the derivatives for bases other than $e$ and inverse trig functions.

  • Exponential (Base $a$): Problem 4 involves $y = 3^{(4x+1)}$. Don't forget the $\ln(3)$ term!
  • Inverse Tangent: Problem 7 requires the rule for $m(x) = \tan^{-1}(5x^2 - 4)$. Recall that $\frac{d}{dx}\tan^{-1}(u) = \frac{u'}{1+u^2}$.

4. Implicit Differentiation

When $x$ and $y$ are mixed together, we cannot simply say $y=...$. You must apply Implicit Differentiation.

  • Problem 6: $$x^2 + 6xy + 12y^2 = 28$$
  • Problem 10: $$3xy = 4x^2 - 4y^2 + 6$$

Crucial Reminder: When deriving the term $6xy$ or $3xy$, you must use the Product Rule. Also, every time you derive a $y$ term, you must attach a $\frac{dy}{dx}$ or $y'$.

5. Applications: Tangent Lines

Finally, we apply the derivative to find the equation of a tangent line at a specific point.

  • Problem 9: $$y = \frac{1+x}{3+e^x} \quad \text{at point } (0, \frac{1}{4})$$

Remember the process:

  1. Find the derivative $y'$ (Slope function).
  2. Plug in the given $x$-value to find the specific slope $m$.
  3. Use the Point-Slope form: $y - y_1 = m(x - x_1)$.

Work through the attached PDF carefully. If you can handle these ten problems, you are in great shape for the exam. Good luck studying!