14. A student did the following. \( \begin{aligned} \frac{d}{d x}(x) & =\frac{d}{d x}(\underbrace{1+1+1+\ldots+1}_{x \text { times }}) \\ & =0+0+0+\ldots+0 \\ & =0\end{aligned} \) What is wrong with this reasoning?

Let \( x y+e^{x}-e^{y}=0 \), then \( y^{\prime}(0)= \)

\( \frac{d}{dx}\frac{1}{sin^{2}x} \)

If \( x^{3}+y^{3}=6 x y \), then find \( \frac{d y}{d x} \) and \( \frac{d^{2} y}{d x^{2}} \)

22. a) If \( f(x)=\frac{|6-x|}{x-6} \), find \( \lim _{x \rightarrow 6+} f(x) \) (if exist.) b) Evaluate: \( \lim _{x \rightarrow 0} \frac{\tan x-\sin x}{x^{3}} \) c) Evaluate: \( \lim _{x \rightarrow 0} \frac{x\left(2^{2}-1\right)}{1-\cos x} \) KMC Note d) Find \( \frac{d x}{d y} \) if \( x^{3}+y^{3}=3 a x^{2} y \) e) What is the derivative of \( \sec ^{-1} x \) with respect to ' \( x \) '.

4. (15б) Выясните, является ли функция \( z=\ln \left(x^{2}+y^{2}+2 y+1\right) \) решением уравнения \( \frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}=0 \)