Intereting Posts

Evaluating $\prod\limits_{n=2}^{\infty}\left(1-\frac{1}{n^3}\right)$
Derivation of inverse sine, what is wrong with this reasoning?
Semisimple Lie algebras are perfect.
Is Category Theory geometric?
For what $n$ is $U_n$ cyclic?
An Integral involving $e^{ax} +1$ and $e^{bx} + 1$
What is the Probability that a Knight stays on chessboard after N hops?
What is intresting about $\sqrt{\log_x{\exp{\sqrt{\log_x{\exp{\sqrt{\log_x{\exp{\cdots}}}}}}}}}=\log_x{e}$?
Proving that $e$ is irrational using these results
Show that is $\int_A f$ exists, then so does $\int_A |f|$
Continuous Function
Do mathematicians, in the end, always agree?
How does a complex power series behave on the boundary of the disc of convergence?
Geometrico-Harmonic Progression
Please help: My MATLAB code for solving a 2D Schrödinger equation keep giving me weird output.

It seems to be the case, but i don’t have a proof.

Given the function $f$ such that $f(x) \geq e^x$, is it true that $f'(x) \geq f(x)$?!

I was experimenting with wolfram and it appears that $\frac{f'(x)}{f(x)} \geq 1$ whenever $f$ is bigger or equal to $\exp(x)$.

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- Differentiation - simple case
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- Academic reference concerning Minkowski's question mark function
- Finding the limit $\lim_{x\rightarrow 0^{+}}\frac{\int_{1}^{+\infty}\frac{e^{-xy}\quad-1}{y^3}dy}{\ln(1+x)}.$
- $\nabla \cdot \color{green}{(\mathbf{F} {\times} \mathbf{G})} $ with Einstein Summation Notation

Note : as suggested in the comments, i meant that for all positive $x$ which means that $f(x) \geq e^x \space \forall x$ such that $x\ge 0$.

- Proving a function is constant, under certain conditions?
- Why is there only one term on the RHS of this chain rule with partial derivatives?
- problem on continuous and differentiable function
- $f$ not differentiable at $(0,0)$ but all directional derivatives exist
- If $\displaystyle \lim _{x\to +\infty}y(x)\in \mathbb R$, then $\lim _{x\to +\infty}y'(x)=0$
- Is the determinant differentiable?
- Product rule for scalar-vector product
- Derivative of a function with respect to another function.
- limits of integration and derivative
- Function whose third derivative is itself.

Let us consider the function $g(x):=\log(f(x))$.

Your question is recast as: if $g(x)\ge x$, is it true that $g'(x)\ge1$ ?*

This is obviously false, as a curve lying above $x$ can take any slope.

*$\log f(x)\ge \log e^x$ vs. $(\log f(x))’=\dfrac{f'(x)}{f(x)}\ge1$.

No and heres why.

Assume u have a function $f(x) = e^x+1$ which is greater than $e^x$ for all $x$.

However its derivative would be $f'(x) = e^x$ which is smaller. I think a more interesting question would be if the function was asymptotically greater than $e^x$

This is not true. The $e^x$ is always positive, so it suffices to give a function that is decreasing on some interval and yet greater than $e^x$. It should be obvious that such functions exist.

let $f(x) = e^x + 1 + \cos x \ge e^x + 1 – 1 = e^x$

$f'(x) = e^x – \sin x$.

If $f(x) \geq e^x \space \forall x$ then we can take the logarithm of both sides to give $\log f(x) \geq x \space \forall x$. It is tempting to take the derivative of both sides and conclude that $\frac{f'(x)}{f(x)} \geq 1 \space \forall x$ – but, as the examples above show, that is just not true.

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- Show that the sequence $1,2,3,4,5,6,7,8,9,1,0,1,1,1,2,1,3,1,4,1,5,1,6,1,7,1,8,1,9,2,0,2,1,\cdots$ isn't periodic
- Let$\ p_n$ be the$\ n$-th prime. Can you give me a proof for$\ \prod_{i=1}^\infty \frac{p_i-1}{p_i}=P\approx \frac{1}{11.0453}$?
- Axiom of Choice and finite sets
- $\epsilon$ – $\delta$ definition of a limit – smaller $\epsilon$ implies smaller $\delta$?