27 votes

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

The root of the difficulty is that $x$ appears free in $f(z)$, but we are trying to "capture" it with $g(x)$, which is illegal. When we substitute $g(x)$ into $f(g(x))$, we have a variable clash: $$ f(...
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  • 556
27 votes
Accepted

How does one tutor an A-level student past the derivative paradox?

There is no royal road to geometry. - Euclid Nor calculus. The essence of calculus thinking is really the limit concept. One needs to wrap one's mind around that. Formally: it's the core technique ...
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26 votes
Accepted

Finding the Balance in a Math Question (Teaching)

I think it is helpful to let students know that you are looking for their thinking while problem-solving, and not just answers. Then you can ask questions like: Find all of the points on a circle of ...
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  • 788
26 votes

How does one tutor an A-level student past the derivative paradox?

I teach calculus at a community college in the U.S. (2 year college, from which many students transfer to a university). I explain limits from about day two in an informal way ("h gets infinitely ...
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  • 17.5k
18 votes

Are there direct practical applications of differentiating natural logarithms?

Have you thought about the fact that you’re asking this in the middle of a pandemic for which log plots are being used all over the place to visualize the growth of COVID cases? At any rate, $${d \...
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  • 4,619
17 votes

What are some examples of great functions that are not too elementary (easy)?

I'm personally a fan of simple examples: $x e^x$ (has nice critical point, point of inflection) $e^{-x^2}$ (with appropriate rescaling, the normal distribution from statistics) $\frac{x}{x^2+1}$ (a ...
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  • 2,089
16 votes

How does one tutor an A-level student past the derivative paradox?

We routinely get questions related to pushing more rigor in early calculus. Usually from outstanding students and based on sample of one I like it that way logic. There's a reason why things are the ...
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  • 177
14 votes

Are there direct practical applications of differentiating natural logarithms?

Whenever we measure a quantity on a log scale (such as Richter, decibels, musical pitch, or a log-plot axis), we are focusing attention on relative variation in that quantity. If $y = \ln x$, we have $...
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  • 271
12 votes

How does one tutor an A-level student past the derivative paradox?

Start with a numerical example. Say you want to find the gradient of the tangent to $y=x^2$ at $x=1$. Obviously the point itself is $(1,1)$. Pick a nearby point, say $x=1.1$. A moment with a ...
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11 votes

A different symbol for the indefinite integral/antiderivative?

First, yes, many teachers use that term, "anti-derivative" and "integral" only when a definite integral is in use. For your examples of each, it seems to me that you offered a clear distinction ...
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10 votes

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

f is not a function of (only) z - f here is a function of x as well as z. I think this explanation is intelligible to a calc 1 student, and gets at the heart of the matter.
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9 votes

Finding the Balance in a Math Question (Teaching)

Note: It's possible that in the future, Wolfram Alpha will improve and be able to answer the questions in this answer, so it's best to actually try them in Wolfram Alpha first. Use questions that ...
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  • 10.2k
9 votes

What are some examples of great functions that are not too elementary (easy)?

Don't forget purely graphical problems. Give a graph of $f$ or $f'$, ask for the student to sketch the other one.
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8 votes

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

As mentioned, a probable cause is an implicit reasoning as if every operation were a homomorphism (similar to the implicit reasoning by linearity). Similar errors include $\ln(x+y) = \ln(x)+\ln(y)$, $...
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8 votes

How does one tutor an A-level student past the derivative paradox?

I admit that I'm unable to follow the proof you give as an example in your question, but am I correct when I assume that your question simply wonders how to reconcile $dy/dx$ with the fact that $dx$ ...
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8 votes

Good exercises that force you to apply the definition of the derivative, without explicitly telling you to do so?

Here are some example questions. The graph of the function $f$ is given above. Evaluate the following limits. If the limit is infinite, write $\infty$ or $-\infty$ as appropriate. If the limit does ...
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8 votes

Do we need to practice equation derivation while learning math if equations will be chunked and automatized?

I endorse Steve's comment about different skills and different levels of proficiency in different skills. Also, if one at one time justified the Pythagorean theorem, but is no longer able to do so, I ...
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7 votes

Introducing derivative concept and definition

It occurs to me that maybe you can approach this from the viewpoint of approximations. Briefly, I’m thinking of the kinds of approximations they likely use in practice, and which are often given in an ...
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7 votes
Accepted

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

$$ \frac{d (3^{5x+1})}{dx} = f'(g(x))g'(x)= \frac{d \left(3^{5x+1}\right)}{d(3)} \times \frac{d (3)}{dx}. $$ However $\dfrac{d (3^{5x+1})}{d(3)}$ is undefined.
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  • 210
7 votes
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When analytic form of derivatives is preferred over numerical form?

I think maybe if you try to write down precisely what you mean by $h<<1$ you will end up writing the definition for $\lim_{h \to 0}$. From this point of view, there is really no difference ...
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  • 19.1k
7 votes

How does one tutor an A-level student past the derivative paradox?

First off - I 100% agree with Collins here that there's no shortcuts. To really understand how derivatives work, you need to learn the epsilon-delta definition. But it's my sense that you're not ...
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7 votes

Good exercises that force you to apply the definition of the derivative, without explicitly telling you to do so?

I'll argue that this will be likely not feasible for a test question. There's several points in the calculus progression where there's a "hard bottleneck" of some sort, but once you get past ...
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6 votes
Accepted

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

Thinking about tangent lines as linear approximations is a great way to foreshadow Taylor series. After linear approximations in calc 1, I usually give students a few prompts to get them thinking ...
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  • 593
6 votes

Are there direct practical applications of differentiating natural logarithms?

I couldn't find a lot either. Suggest playing with some logarithmic properties and constructing problems based on that. E.g. pH is log10 of the hydronium ion concentration. Could ask how the pH ...
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  • 149
6 votes

Good exercises that force you to apply the definition of the derivative, without explicitly telling you to do so?

Give a function together with some properties of the function, but do not give a formula, and ask for the derivative. An example: Let $f: \mathbb{R} \to \mathbb{R}$ be a function which enjoys the ...
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5 votes

Are there direct practical applications of differentiating natural logarithms?

A couple of direct applications: Showing that a power law appears on a log-log graph as a straight line with gradient equal to the exponent of the power law (although that can be done by other, ...
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5 votes

Are there direct practical applications of differentiating natural logarithms?

Boltzmann's equation for entropy is $S=k\ln W$, and the second law of thermodynamics is all about change in entropy. Maybe this is a place to start with your quest for a practical application of the ...
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  • 7,703
5 votes

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

The main reason we have the word "critical point" is for the first derivative test. The statement is slightly easier if you only have to say "critical point" instead of "...
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5 votes

A different symbol for the indefinite integral/antiderivative?

Before inventing new notation, it is very important to learn the accepted notation and teach it to your students correctly. The question contains the following claim: $$\int \cos = \sin$$ which is ...
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  • 19.1k
5 votes

A different symbol for the indefinite integral/antiderivative?

I'll echo other responses with the same: Do NOT introduce made up notation. I've made the effort in my Calculus courses to follow your outline while avoiding any new symbols. Similar to user20311's ...
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