# Questions tagged [calculus]

For basic questions about limits, derivatives, integrals and applications, mainly of one-variable functions.

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### If $f(x)$ is integrable then $f(e^{x})$ is also integrable on $\Bbb R^{+}$ [on hold]

If $f(x)$ is integrable then is it true that $f(e^{x})$ is also integrable on $\Bbb R^{+}$?
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### Derivative of trace of a product containing an inverse matrix

What's the derivative of $$f(X)=\text{Tr}(YX^{-1})$$ with respect to $X$, where $X$ and $Y$ are square matrices of the same dimension? My first attempt is to apply the chain rule as: Let $h(X)=X^{-1}$...
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### How to evaluate the following derivative of integral function [on hold]

$$\frac{d}{dt}\left(\int_{0}^{t}\frac{1 - e^{a(t-x)}\operatorname{erfc}\left(\sqrt{a(t-x)}\right)}{\sqrt{x}(x+b)}\,dx\right)$$
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### How are derivatives and integrals exempt from the chain rule? [on hold]

A question I never considered when studying calculus is why the chain rule seems to fail when applied to derivatives and indefinite integrals. For example, according to the chain rule, the derivative ...
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### Why does integration have to be the reverse operation of differentiation? [duplicate]

I am in High school and when I was learning calculus, we were taught that integration is nothing but the reverse of differentiation. But I really don't get it why does that needs to be the case. ...
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### Explicit expression for $1/e$ as a limit

I am following an elementary math book: What is Mathematics and currently referring to infinite series representation of the exponent. In deriving the explicit formula for $e$ and $1/e$, the author ...
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### Argmin using differentiation with respect to inverse matrix

I'm trying to understand the following step in a calculation: My problems: (1) If we want the argmin with respect to $R$, why are we not differentiating with respect to $R$? I assume differentiating ...
1answer
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### Product rule: why not $δ_{xy}=xδ_y+yδ_x+δ_xδ_y$?

I'm reading V.I. Arnold's Huygens and Barrow, Newton and Hooke and he gives this diagram as part of his discussion of the derivation of the product rule, and I can't see what's wrong with it, except ...
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### Solving $\int \frac{e^{2x}}{1+e^x} \, dx$

Problem: $\int \frac{e^{2x}}{1+e^x} \, dx$ My book says to divide in order to solve by getting $\int e^x-\frac{e^{x}}{1+e^x} \, dx$ but how am I supposed to divide? I tried long division but ...
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### Example of a function that is non-differentiable on an open interval

I am studying a course on single variable calculus. During a lecture, the professor mentioned in passing that there can be functions that are non-differentiable on an entire open interval. Can anyone ...
1answer
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### Integrals Definite substitution from this? [on hold]

Integrals Definite substitution from $$\int^{\pi/4}_{0}\frac{8 \cos(2t)}{\sqrt {9-5 \sin t (2t)}}~dt ~?$$? u = 9 - 5 sin t(2t) du = -10(t cos (t) + sin (t)) , this correct for du = -10(t cos (t) ...
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### reparametrization function

I am trying to understand some computation, but it seems something is not going well. You will find this slide in the following link: https://www.asc.ohio-state.edu/kurtek.1/Lecture3_Srivastava.pdf ...
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### From binomial theorem to differential calculus

I first posted this over at HSM, without much uptake. I'm trying to understand the development of the calculus. Does this sound plausible as one of the stages? Newton knows the binomial theorem, ...