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?

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

Let \( y-x e^{y}=1 \), then \( d y(0)= \)

a) \( D[(1-x) \sqrt{1+x}] \)

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 ' \( x \) '.

Ex. 1. A company manufactures and sales \( X \) units of radio transistor per week. If the weekly cost and demand equations are: \[ C(X)=5,000+2 x \] \[ P(X)=10-\frac{x}{1,000} \quad P \text { is price } \quad[0,8000] \] or each week find:- 1) The production level that leads to maximum revenue and the maximum revenue. 2) The production levels that leads to maximum profit and the maximum profit. 3) The production level that leads to minimum cost and the minimum cost.