# 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 ...
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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 ...
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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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