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 ...
2
votes
1answer
151 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 $...
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
353 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 ...
4
votes
2answers
164 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 ...
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 ...
3
votes
1answer
77 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 ...
1
vote
1answer
172 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 $$f^{(n)}(x)=y^{(n)}$$
6
votes
2answers
282 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 ...