New posts in Derivatives

Derivative when function approaches infinity.

Show that $f(x)=\sqrt 2$ has no solutions $f(x)=\sin x\cos x(2+\sin x)$ and $x\in [0,\frac{\pi}{2}]$.

Prove that $\ln(1+\frac{1}{x}) < \frac{1}{({x^2 + x})^{1/2}}$

Gradient of $\mbox{dist}\left(x, D \right)^2:= \left\| x - P_{D}(x)\right\|_2^2$, where $P_{D}(x)$ is a projection operator

Show that $\exp\left(\frac{1}{x}\log\frac{e^{x}-1}{x}\right)$ is increasing.

Is there a real-analytic monotone function $f:(0,\infty) \to \mathbb{R}$ which vanishes at infinity, but whose derivative admits no limit?

Derivative of a unit vector

Constant function composition

$f''(x) = g(x)$ and $g''(x) = f(x).$ Suppose also that $f(x)g(x)$ is linear in $x$ on $(a,b).$ Show that $f(x) = g(x) = 0$ for all $x ∈ (a,b).$

Proof using Fundamental Theorem of Calculus (Showing RHS = LHS)