Professor Baker's Math Class: Fall 2025 Chapter 3 Test Review

Welcome, students! This review is designed to help you prepare for the upcoming Chapter 3 test. We'll be focusing on derivatives and their applications. Remember, practice is key to mastering these concepts. Let's get started!

Key Concepts

  • Power Rule: If $y = x^n$, then $\frac{dy}{dx} = nx^{n-1}$.
  • Constant Multiple Rule: If $y = cf(x)$, then $\frac{dy}{dx} = c \frac{d}{dx}f(x)$.
  • Sum/Difference Rule: If $y = u(x) \pm v(x)$, then $\frac{dy}{dx} = \frac{du}{dx} \pm \frac{dv}{dx}$.
  • Product Rule: If $y = u(x)v(x)$, then $\frac{dy}{dx} = u'v + uv'$.
  • Quotient Rule: If $y = \frac{u(x)}{v(x)}$, then $\frac{dy}{dx} = \frac{u'v - uv'}{v^2}$.
  • Chain Rule: If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x))g'(x)$.

Example Problems

Let's work through some examples similar to what you might see on the test:

Problem 1: Applying the Power Rule and Constant Multiple Rule

Find the derivative of $y = \frac{1}{5x^5} - x^2$.

First, rewrite the function as $y = \frac{1}{5}x^{-5} - x^2$.

Now, apply the power rule and constant multiple rule: $\frac{dy}{dx} = \frac{1}{5}(-5)x^{-6} - 2x = -x^{-6} - 2x = -\frac{1}{x^6} - 2x$.

Problem 2: Derivatives with Radicals and Negative Exponents

Find the derivative of $y = \sqrt[3]{x} + \frac{1}{x^5}$.

Rewrite as $y = x^{\frac{1}{3}} + x^{-5}$.

Applying the power rule: $\frac{dy}{dx} = \frac{1}{3}x^{-\frac{2}{3}} - 5x^{-6} = \frac{1}{3\sqrt[3]{x^2}} - \frac{5}{x^6}$.

Problem 3: Product Rule

Find the derivative of $y = (6z^2 + 3)e^z$.

Using the product rule, let $u = 6z^2 + 3$ and $v = e^z$. Then $u' = 12z$ and $v' = e^z$.

$\frac{dy}{dz} = (12z)e^z + (6z^2 + 3)e^z = e^z(6z^2 + 12z + 3)$.

Problem 4: Quotient Rule

Find the derivative of $g(x) = \frac{7 - x^4}{5e^x}$.

Using the quotient rule, let $u = 7 - x^4$ and $v = 5e^x$. Then $u' = -4x^3$ and $v' = 5e^x$.

$g'(x) = \frac{(-4x^3)(5e^x) - (7-x^4)(5e^x)}{(5e^x)^2} = \frac{-20x^3e^x - 35e^x + 5x^4e^x}{25e^{2x}} = \frac{-4x^3 - 7 + x^4}{5e^x}$.

Remember to review implicit differentiation, related rates, and optimization problems as well. Good luck with your test! You've got this!