[/quote]
,
,
.
lies on 
such that 
.
passing trough 
at 
.
.
such that
,
.
and 
the midpoints of 
and 
.
and 
meet 
and 
at 
and 
,
,
bisects both angles 
,
,
.
be the circumcircle of triangle 
,
be the tangent line of 
passing 
.
be the points each on side 
such that 
.
meets 
.
meets 
passing 
.
and tangent to line 
.
![\[
\text{Find all prime numbers } (p, q) \text{ such that } p^2 + 3pq + q^2 \text{ is a fifth power of an integer.}
\]](http://latex.artofproblemsolving.com/c/f/8/cf855cb087102024a50d41f49c67e7fd1ef22767.png)
be an integer, and let 
be positive integers. Show that there exist positive integers 
satisfying the following three conditions:
for 
the remainders of 
are pairwise different; and
denotes the integer part of real number 
,
.
.
;
, ∀x∈[0,1].
Y by CXNJ.AoPS, At777
Hello, everybody!
This is a differentiation marathon. It is just like an ordinary marathon, where you can post problems and provide solutions to the problem posted by the previous user. You can only post differentiation problems (not including integration and differential equations) and please don't make it too hard!
Have fun!
(Sorry about the bad english)
This is a differentiation marathon. It is just like an ordinary marathon, where you can post problems and provide solutions to the problem posted by the previous user. You can only post differentiation problems (not including integration and differential equations) and please don't make it too hard!
Have fun!
(Sorry about the bad english)
This post has been edited 5 times. Last edited by LawofCosine, Mar 11, 2025, 10:30 PM
requires the chain rule, so we let 
and 
.
:
So, we get![\[\frac{d}{dx}((1+e^x)^x)=x\cdot e^x\cdot (1+e^x)^{x-1}+\ln(1+e^x)\cdot (1+e^x)^x.\]](http://latex.artofproblemsolving.com/0/2/e/02e5a05a7a65c046385dea38fa5467f3a38a0164.png)
![\[\frac{d}{dx}\left(\frac{x^2}{(1+e^x)^x}\right)=\boxed{\frac{2x\cdot(1+e^x)^x-x^2(x\cdot e^x\cdot (1+e^x)^{x-1}+\ln(1+e^x)\cdot (1+e^x)^x)}{(1+e^x)^{2x}}}.\]](http://latex.artofproblemsolving.com/2/e/8/2e8646ac8e1c25ce32c05836e24d98c18314d87a.png)
![\[\frac{d}{dx}\left(\frac{e^x}{x^2}\right)\]](http://latex.artofproblemsolving.com/4/b/9/4b9dad962e9007e8e48ec896cb617157513364ca.png)
with respect to 
a
(for 
)
in function of 
(for 
.
.
![$$\frac{\rm d\it W(x)}{\rm d\it x}=\frac{1}{\frac{\rm d\it x}{\rm d\it W(x)}}=\frac{1}{\frac{\rm d\it[W(x)e^{W(x)}]}{\rm d\it W(x)}}=\frac{1}{[W(x)+1]e^{W(x)}}=\frac{1}{x+e^{W(x)}}$$](http://latex.artofproblemsolving.com/8/3/7/8373d414a0a4c274af22f7e3e3b261a68e56eb29.png)
![\[I=\int\limits _0^{b^2}\left(ax+b\sqrt x+c+\frac{b}{\sqrt x}+\frac{e}x\right)\,\text dx,\]](http://latex.artofproblemsolving.com/6/6/4/6643181ee662e77234c9dff892b1e23a2d224f0a.png)
we apply linearity, and get ![\[I=a\int\limits _0^{b^2}x\,\text dx+b\int\limits _0^{b^2}\sqrt x\,\text dx+c\int\limits _0^{b^2}\text dx+b\int\limits _0^{b^2}\frac1{\sqrt x}\,\text dx+e\int\limits _0^{b^2}\frac1x\,\text dx.\]](http://latex.artofproblemsolving.com/6/d/e/6de8cccf906bef4472120a527390b69016edbc0c.png)
This integral diverges to infinity because the final integral is divergent unless 
So, ![\[\boxed{I=\infty\,\text{sgn}(e)},\quad e\ne0,\]](http://latex.artofproblemsolving.com/f/5/9/f5945fc7b2e39e6e8c1f9dc6ddd54b26c476bf61.png)
or ![\[\boxed{I=\frac{ab^2}2+\frac{2b^3|b|}3+cb^2+2b|b|,}\quad e=0.\]](http://latex.artofproblemsolving.com/0/b/6/0b66e3c47b341c2ba832127ef380c6eac6215d8d.png)
and 
as 
Evaluate ![\[(f*g)(t).\]](http://latex.artofproblemsolving.com/0/4/4/044b43fa6f4112b89cf81c71e17cd9587da4c479.png)
![\[(f*g)(t)=\int\limits _{-\infty}^{\infty}f(\tau)g(t-\tau)\,\text d\tau.\]](http://latex.artofproblemsolving.com/6/2/3/6231f7185479d8e27f4735eee1a9db242c19bf5a.png)
It is commutative, distributive, linear, and associative.
![$[e^{-e}, e^{\frac 1e}]$](http://latex.artofproblemsolving.com/a/4/2/a42f578224ffb9b289df8222619802ee91c254d8.png)
.
.
,
.
and a point 
.
,
the tangent line on 
and 
the normal line on 
such that the tangent line 
at the point 
of the parabola is perpendicular to 
of the parabola, 
,
.
and find, if posible, the minimum area and when is obtained.
with the points 
and 
.
and 
;
.
.
,
.
.
,
.
.
