A function satisfies the equation $f(x+y) = f(x) f(y)$. Show that $f(x)$ is differentiable at all $x$, and determine its $f'(x)$ in terms of $f(x)$.
138864
If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ and $f'(0)$ exists, then $f$ is differentiable on $\mathbb R$
69707
If $f(x + y) \leq yf(x) + f(f(x))$ for all real numbers $x$ and $y$, then prove that $f(x) = 0$ for all $x ≤ 0$.
97146
differentiate with respect to a function
23465
When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?
83707
Suppose that $f$ is differentiable on $\mathbb{R}$ and $\lim_{x\to \infty}f'(x)=M$. Show that $\lim_{x\to \infty}f(x+1)-f(x)$ exists and find it.
143491
Show that $f$ is differentiable on $(−1, 1)$.
58943
If $f'(x)\le g'(x)$, prove $f(x)\le g(x)$
112923
Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ a differentiable function such that $f'(x)=0$ for all $x\in\mathbb{Q}$
105068
Proving that the function is differentiable and $f'(0)=0$
49705
Is $f(x,y)=(x+2y+y^2+|xy|, 2x+y+x^2+|xy|)$ differentiable?
104278
Finding function $f(x)$ which satisfy given functional equation
71127
How do I find $f(0)$, $f'(0)$, and $f'(x)$ given $f(x+y)=f(x)+f(y)+x^2y+xy^2$ and $\lim_{x\to0}\frac{f(x)}{x}=1$?
78834
if f(x) if differentiable and continuous, prove $\frac{af(a)-bf(b)}{a-b} = f(c) + cf'(c) $
111458
Prove that $f$ is differentiable
85480
Let $f(x,y)=\frac{x^3-y^3}{x^2+y^2}$. Is f differentiable in $(0,0)$?
125788
A function with a property $f(x+y)=f(x)f(y)$
88683
Functional equation $f\left(\frac{x+y}{2}\right)+f\left(\frac{2xy}{x+y}\right)=f(x)+f(y)$ implies $2f(\sqrt{xy})=f(x)+f(y)$