Gateway Review Activity

[greybox2] Last updated March 7th @ 7:00p: I fixed a couple of typos, added hints for what errors to look for, and made some stylistic changes so it is easier to navigate the page. Otherwise, the activity and problems are exactly the same as the ones we did in class. You may email me if you need further hints, clarification, or any kind of help. :) [/greybox2]

When taking derivatives, it is easy to make a variety of mistakes (some of which you may not even realize). This review is intended to try to get you to discover some of these mistakes for yourself. Below, you will find some problems I have selected from Dr. Browne’s Gateway problems list.pdf file and wrote up bogus solutions to that are plagued with mistakes. In your groups, you will discuss the following:

  1. Identify the mistake(s) (if any) that are present.
  2. Correct the mistake(s) by writing down the correct solution (and also the work).

I will come around to each group and check your progress periodically. Notice I did not give the answers to the problems. This is intentional because you need to be able to know how to double-check your own work without relying on something to guide you (this is the unfortunate environment we all face in exams).

That being said, if you want to check your answer after you worked everything out and I’m busy helping another group, I would recommend using WolframAlpha. Here is an example of how to use it:

https://www.wolframalpha.com/input?i=differentiate+%281+%2B+x%29%2F%28x%5E2+%2B+1%29

If you use ChatGPT or DeepSeek for double-checking your work, be extra careful — it’s usually pretty good but will make hard-to-spot computation or logic errors.

Tricky problem to be careful about

[graybox] If Problem 9 pops up in the gateway, just leave the answer as $2\dfrac{2}{3} \cdot 3x^2 + 2x + 12$ and do not attempt to simplify anything. [/graybox]

Problem 20

[graybox] Problem: Differentiate

\[\begin{align*} p(x) = 16x^3 + \dfrac{17}{\sqrt{x}} - 10x^{3.1416} + \pi^2. \end{align*}\]

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[graybox2] Bogus solution to critique: We begin by rewriting $p(x)$ as

\[\begin{align*} p(x) = 16x^3 + 17x^{-1/2} - 10x^{3.1416} + \pi^2. \end{align*}\]

Then compute the derivative as

\[\begin{align*} \frac{d}{dx} = 48x^2 + \frac{17}{2}x^{-3/2} - 31.416x^{2.1416} + 2\pi. \end{align*}\]

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[hint] There are three mistakes.

  • Sign error.
  • Incorrect application of a derivative rule. If you cannot find this one, you need to review your derivative rules. Also recall that $\pi$ is a number/constant, not a variable.
  • Incorrect notation. If you cannot find it, you need to review what the notation $\dfrac{d}{dx}$ means. [/hint]

Problem 36

[graybox] Problem: Differentiate

\[\begin{align*} g(y) = (\sqrt{y} - 2)(1 - y^2). \end{align*}\]

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[graybox2] Bogus solution to critique: We note that $g(y)$ is the product of two functions and so we apply the product rule. This gives

\[\begin{align*} \frac{d}{dx}[g(y)] &= \frac{d}{dx}[(\sqrt{y} - 2)]\cdot \frac{d}{dx}[(1 - y^2)] \\ &= \frac{d}{dx}[(y^{1/2} - 2)]\cdot \frac{d}{dx}[(1 - y^2)] \\ &= \frac{1}{2}y^{-1/2} \cdot (-2y) \\ &= -y^{1/2}. \end{align*}\]

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[hint] There are two types of mistakes here:

  • Incorrect application of derivative rule. If you cannot find it, you need to review your derivative rules.
  • Incorrect notation. If you cannot find them, you need to review what the notation $\dfrac{d}{dx}$ means and possibly read more carefully. [/hint]

Problem 52

[graybox] Problem: Differentiate

\[\begin{align*} r(u) = \frac{5 + u^2}{1 - u^3}. \end{align*}\]

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[greybox2] Bogus solution to critique: Since $r(u)$ is a fraction of functions, we apply the quotient rule. Doing so gives

\[\begin{align*} r(x) &= \frac{\dfrac{d}{dx}[(1 - u^3)] \cdot (5 + u^2) - \dfrac{d}{dx}[(5 + u^2)] \cdot (1 - u^3)}{(1 - u^3)^2} \\ &= \frac{-3u^2 \cdot (5 + u^2) - 2u \cdot (1 - u^3)}{(1 - u^3)^2}. \end{align*}\]

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[hint] There are two types of mistakes here:

  • Incorrect application of a derivative rule. If you cannot find it, you need to review your derivative rules.
  • Incorrect notation. If you cannot find them, you need to read more carefully and/or review what the notation $\dfrac{d}{dx}$ means. [/hint]

Problem 66

[graybox] Problem: Differentiate

\[\begin{align*} h(w) = (1 + \sqrt{w^3 + 3})^4. \end{align*}\]

[/graybox]

[graybox2] Bogus solution to critique: We can realize $h(w)$ as the composition of an “outside” function and an “inside” function. Thus, we apply the chain rule to obtain

\[\begin{align*} h'(w) = 4(1 + \sqrt{w^3 + 3})^3 \cdot \frac{d}{dw}[(1 + \sqrt{w^3 + 3})]. \end{align*}\]

To compute the derivative of $1 + \sqrt{w^3 + 3}$, we first rewrite then differentiate:

\[\begin{align*} \frac{d}{dw}[(1 + \sqrt{w^3 + 3})] &= \frac{d}{dw}[(1 + (w^3 + 3)^{1/2})] \\ &= \frac{d}{dw}[(w^3 + 3)^{1/2}]. \end{align*}\]

Since $(w^3 + 3)^{1/2}$ is a composition of another “outside” function and an “inside” function, we apply chain rule again to get

\[\begin{align*} \frac{d}{dw}[(w^3 + 3)^{1/2}] = \frac{1}{2}(w^3 + 3)^{-1/2}. \end{align*}\]

Thus, putting everything back together, we get

\[\begin{align*} h'(w) &= 4(1 + \sqrt{w^3 + 3})^3 \cdot \frac{1}{2}(w^3 + 3)^{-1/2} \\ &= 2(1 + \sqrt{w^3 + 3})^3 \cdot (w^3 + 3)^{-1/2}. \end{align*}\]

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[hint] There is one mistake here.

  • Incorrect application of a derivative rule. If you cannot find it, you need to review your derivative rules. [/hint]

Extra thing to think about

If you made it past the four problems above, then try to answer the following:

[greybox2] What do you think are good strategies to evaluate derivatives of very complicated functions? [/greybox2]