Questions tagged [derivative]

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16
votes
6answers
2k views

Are there direct practical applications of differentiating natural logarithms?

The textbook I am using to teach Calculus I includes in the exercises of most chapters a number of interesting real-world applications of the concepts from that chapter. However, the chapter on the ...
13
votes
3answers
2k views

Finding the Balance in a Math Question (Teaching)

As we try to work and teach in the midst of this pandemic, some problems arise when making online math exams. My question is simple: What could be an interesting basic differentiation question such ...
12
votes
9answers
3k views

How to explain what's wrong with this application of the chain rule?

Yesterday a student in my calculus class attempted something like this: Problem statement: Find the derivative of $3^{(5x+1)}$ with respect to $x$. Proposed solution: Let the inner function be ...
11
votes
5answers
1k views

A different symbol for the indefinite integral/antiderivative?

Examples. An indefinite integral (or antiderivative) of $\cos$ is $\sin$: $$\int \cos = \sin.$$ Edit: There has been much unexpected confusion with the above statement. I define the above statement ...
6
votes
4answers
376 views

Ideas for the introduction of the derivative?

I want to introduce to my class to the derivative, but I am still searching for a good, realistic context that isn't too hard to understand, without seeming to be contrived. Do you have an ideas for ...
6
votes
2answers
284 views

Why most people think that :$(fg)'=f' \cdot g'$?

let $f$ and $g $ be two real valued function , I have asked many students what is the derivative of $(fg)'$ they answered me :it is $f' \cdot g'$, then I seek why most people (students) guess that ?
4
votes
3answers
2k views

When analytic form of derivatives is preferred over numerical form?

Is there a specific example when the analytic form of a derivative $\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$ is preferred to the numerical form $\frac{f(x+h)-f(x)}{h}$, $h \ll 1$? Are there cases when the ...
4
votes
2answers
167 views

Introducing derivative concept and definition

I need to give a short presentation on introducing a class of engineering students to the concept and definition of the derivative. I'm to assume that the students are currently at the appropriate ...
4
votes
0answers
104 views

Term for candidates for inflection points

The critical points of a function $f(x)$ are candidates for local extrema, i.e., if a function changes from increasing to decreasing, or vice versa, it must happen at a critical point. Is there an ...
3
votes
1answer
80 views

Is there a point at which it makes decidedly more sense to learn about a “linear approximation” to a function, rather than a “tangent”?

I'm tutoring a first-semester calculus student, and we were looking over the slides the teacher has used. After teaching (or rather, repeating, for those who completed AP high school math) basic ...
2
votes
1answer
158 views

In single variable calculus, do you distinguish between critical and singular points?

In some texts, a critical point is when the derivative exists and is zero, and a singular point is when the derivative does not exist. So I suppose, at $x=0$, $|x|$ would have a singular point while $...
1
vote
1answer
177 views

Can $y^{(n)}$ be used as a way of representing higher order derivatives?

I have never seen this notation, but I think that it follows in a similar vein for function notation. So if $y=f(x)$, then $y''=f''(x)$. Then by that, can we say that $$f^{(n)}(x)=y^{(n)}$$