Derivative Notation Review
Why the hell am I making you do this?
[greybox2] The next two new topics (implicit differentiation and related rates) really need you to know the ins and outs of Leibniz notation (the $\dfrac{d}{dx}$ notation). Until I say otherwise, I want you to spend the rest of the week using this notation and not the prime notation (e.g. $f’(x)$). [/greybox2]
How to read Leibniz notation
[greybox2] Definition. If $f$ is a differentiable function depending on the variable $x$, then the notation
\[\begin{align*} \frac{d}{dx}f(x) \quad\text{or}\quad \frac{df(x)}{dx} \quad\text{or}\quad \frac{df}{dx} \end{align*}\]is read as “the derivative of $f$ with respect to the variable $x$”. [/greybox2]
Keep in mind that the variable does NOT have to be $x$. For instance, our function could be $f(t)$ in which case the notation needs to be rewritten as
\[\begin{align*} \frac{d}{dt}f(t) \end{align*}\]is to be read as “the derivative of $f$ with respect to the variable $t$”.
[greybox2] From now on, I want you to read the notation in this exact way, adjusting accordingly for the correct variable if needed.
I know this is a mouthful, but you really need to know what you are doing with the notation before you start on implicit differentiation. [/greybox2]
Test your understanding
[greybox] Example 1. Rewrite the basic rules of differentiation, product rule, and quotient rule using Leibniz notation. Don’t bother with the chain rule at the moment (see the below). [/greybox]
[greybox] Example 2. Suppose a problem asks you to differentiate the function
\[\begin{align*} g(t) = t^2 - 3t^{-3} + 5. \end{align*}\]Explain why it is nonsense to say that the derivative of $g$ is
\[\begin{align*} \frac{d}{dx}g(t) = 2t + 9t^{-4}. \end{align*}\]Rectify the mistake and write the correct statement. If you need a hint, click the below box. [/greybox]
[hint] Read, out loud, what the notation $\dfrac{d}{dx}g(t)$ means. What do you notice? If you said something along the lines of “the derivative of $g(t)$” and left it at that, you read the notation wrong and need to reread what I wrote above. [/hint]
What about the chain rule?
If you need to use chain rule to carry out a derivative, then use the old formula with the primes that you are already used to and rewrite the final answer in terms of Leibniz notation if necessary. When we move onto the topic of implicit differentiation, you will see examples of what I mean by this.
If you finish early
If you finish everything above early, I would like you to go over the Gateway Review Activity that we did last Friday if you did not do it already. From here on out in the course, the material will start to build off of the assumption that you know how to take derivatives so you should make sure you really know what you are doing with derivatives.