Let $f$ be a differentiable function such that it satisfies $\int_1^{xy} f(t)dt=x\int_1^{y} f(t)dt+y\int_1^{x} f(t)dt$ then find $f(e)$
86367
Computing $f(x)$ if $\int_1^{xy} f(t) dt = y \int_1^x f(t) dt + x\int_1^y f(t) dt$ and $f(1) = 3$
116946
Ordinary differential equation $y'(t)=\sin(f(t,y))$
89537
$ \int_0^x f(t)dt=\int_0^{ax}f(t)dt+ \int_0^{bx}f(t)dt$ implies $f$ constant
154612
$\int_{0}^{x} f(t) dt \ge {f(x)}$ holds or not on $[0,1]$
31732
If the function $f$ satisfies the equation $f(xf(y)+x)=xy+f(x)$, find $f$
137527
if $f(x)$ is differentiable at a x, prove that: $\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}$
62110
Proving a function $\frac{1}{2y}\int_{x-y}^{x+y} f(t) dt=f(x)$ is a linear polynomial
57695
Show $\inf_f\int_0^1|f'(x)-f(x)|dx=1/e$ for continuously differentiable functions with $f(0)=0$, $f(1)=1$.
37303
Differential of a smooth function
2717
Help proving existence of an specific $x_0$ for which $f''(x_0)=0$
133682
Prove that if $f(t)$ satisfies $f(t)=f'(t)$, then $f(t+s)=f(t)f(s)$ for all $t,s\in R$ without using the fact that $f(t)=c\cdot e^t$.
75932
Differentiability of $g:=f(\sqrt{x^2+y^2})$ for a $C^1$ function with $f'(0)=0$ NBHM $2008$
94193
determine whether $f(x, y) = \frac{xy^3}{x^2 + y^4}$ is differentiable at $(0, 0)$.
56958
bounding a $C^1$ function that upholds $f'(t)\leq 1-f(t)^2$
124346
Where is $f(z)=\Re (z)$ differentiable?
99917
What function satisfies $F'(x) = F(2x)$?
102764
Let $f(x)=\int_0^1|t-x|t~dt$ for all real $x$. Sketch the graph of $f(x)$, what is the minimum value of $f(x)$