1. Week 0: Mild Review
Table of Contents | 2. Week 1: Riemann Sums and Summation Notation ≫Welcome or welcome back to UCLA! Since the week 0 discussion occurs before any lectures, we’ll begin by reviewing the concept of a derivative from several different viewpoints that we will return to in the near future.
Question 1.
Suppose \(f(x)\) is a typical, run-of-the-mill function. What does \(f'(2)\) mean? (Please be more explicit than “the derivative of \(f\) at \(2\)”.)
I am leaving this intentionally open-ended, and there are several different (and equally valid!) ways that one can answer this. Here are a few that I am looking out for:
- \(f'(2) = \lim _{x\to 2} \frac{f(x) - f(2)}{x-2}\). This is the mathematical definition of what \(f'(2)\) is.
- If one drew the graph of \(f\), then drew the line tangent to the graph at the point \((2, f(2))\), \(f'(2)\) would be the slope of that tangent line.
- If \(x\) represented time (in hours since midnight, say) and \(f(x)\) is some measured quantity that changes over time, \(f'(2)\) is the rate at which \(f\) is changing at \(2\) in the morning.
The point is ultimately that there are multiple equivalent ways of interpreting the number \(f'(2)\). When using the mathematical formula, we can apply the product rule, chain rule, quotient rule, and other identitie to compute it. On the other hand, we can use the second geometric interpretation or the third “rate of change” interpretation to use the symbol \(f'(2)\) to represent some real-life quantity.
Let us now return to the problem of computing derivatives. The purpose of the following exercise is to refresh your memory of a few derivative rules and derivatives of common functions.
Problem 2.
Compute the derivatives of the following functions:
- \(e ^{7x}\).
- \((3t-8)\cos \left( 3-t^2 \right)\).
- \(\ln \left( 6x^3-5x+2 \right)\).
- \(\tan x\).
- \(\sin ^{-1}(x+2)\) (difficult!).
Solutions
First, \(\frac{d}{dx} e ^{7x} = 7 e ^{7x}\). This is making sure you remember the chain rule and the derivatives of exponential functions.
The second one needs more work:
\[ \begin{align*} & \frac{d}{dt} \left[ (3t-8) \cos\left( 3-t^2 \right) \right] \\ &= \left[ \frac{d}{dt} (3t-8)\right] \cos \left( 3-t^2 \right) + (3t-8) \left[ \frac{d}{dt}\cos \left( 3-t^2 \right) \right] \\ &= 3 \cos \left( 3-t^2 \right) - (3t-8) \sin \left( 3-t^2 \right) \cdot 2t. \end{align*} \]
One needs to first remember to use the product rule,
then apply the chain rule to the \(\cos \left( 3-t^2 \right)\).
Of course, one needs to remember the derivatives
of trig functions and polynomials.
The third problem is just applying the chain rule, recalling that \(\frac{d}{dx}\ln x = \frac{1}{x}\):
\[ \begin{align*} \frac{d}{dx}\ln \left( 6x^3 -5x+2 \right) &= \frac{1}{6x^3-5x+2} \cdot \frac{d}{dx}\left( 6x^3-5x+2 \right) \\ &= \frac{8x^2-5}{6x^3-5x+2}. \end{align*} \]If you can get the second part, you were probably able to get the third part too.
For the fourth, you either need to have the derivative of \(\tan\) memorised, or you can express \(\tan\) in terms of \(\cos\) and \(\sin\) and apply the quotient rule.
\[ \begin{align*} \frac{d}{dx} \tan x &= \frac{d}{dx} \frac{\sin x}{\cos x} \\ &= \frac{\left( \frac{d}{dx}\sin x \right)\cos x - \sin x \left( \frac{d}{dx} \cos x \right)}{\cos^2 x} \\ &= \frac{\cos^2x + \sin^2}{\cos^2 x} \\ &= \frac{1}{\cos^2 x} = \sec^2 x. \end{align*} \]Note also that we have used the identity \(\sin^2x + \cos^2x = 1\) to simplify at the end!
Finally, the fifth problem can be resolved using implicit differentiation. We first let \(f(x) = \sin ^{-1}(x + 2)\); our goal is to find \(f'(x)\). We have \(\sin \left( f(x) \right) = x+2\), so differentiating both sides and using the chain rule on the left yields
\[\cos \left( f(x) \right) f'(x) = 1.\]Using \(\cos^2(f(x)) + \sin^2(f(x)) = 1\) and \(\sin(f(x)) = x+2\) again, we get \(\cos(f(x)) = \sqrt{ 1 - (x+2)^2}\). Putting it all together, we get
\[\frac{d}{dx} \sin ^{-1}(x+2) = f'(x) = \frac{1}{\sqrt{1 - (x+2)^2}.}\]Do not worry if the fifth solution eluded you; I hope at least that the concept of implicit differentiation looks familiar. If any of these problems were difficult or if you got any of them wrong, I hope that this is a good reminder to review one or more of the following:
- The product rule, chain rule, and/or the quotient rule. It is critical that you remember how and when to use these!
- Basic derivative identities, e.g. of the trig functions.
- Trigonometric identities, such as \(\tan x = \frac{\sin x}{\cos x}\) and \(\sin^2x +\cos^2x = 1\). These will become very, very important when we talk about trigonometric substitutions!