\({dy\over dx}=3x^2+10x+3.\)

The greatest integer function \(f(x)=\lfloor x\rfloor\) is not continuous at every integer point n, since \(\left.\lim _{x \rightarrow n^{-}} \mid x\right\rfloor=n-1\) and \(\lim _{x \rightarrow n^{+}}\lfloor x\rfloor=n .\) Thus f'(n) does not exist.

y = Cosec x . cot x
\(y^{\prime}=\operatorname{cosec} x \frac{d}{d x}(\cot x)+\cot x \frac{d}{d x}(\operatorname{cosec} x)\)
\(=\operatorname{cosec} x\left(-\operatorname{cosec}^2 x\right)+\cot x(-\operatorname{cosec} x \cot x)\)
\(=-\operatorname{cosec}^3 x-\operatorname{cosec} x \cot ^2 x\)
\(=\frac{1}{\sin ^3 x}-\frac{1}{\sin x} \cdot \frac{\cos ^2 x}{\sin ^2 x}\)
\(=\frac{-1}{\sin ^3 x}-\frac{\cos ^2 x}{\sin ^3 x}=\frac{-1-\cos ^2 x}{\sin ^3 x}\)

\(=\frac{-\left(1+\cos ^2 x\right)}{\sin ^3 x}\)

\(f(x)=|x-4|= \begin{cases}-(x-4) ; & ; x<4 \\ (x-4) & ; x \geq 4\end{cases} \)

\(f^{\prime}(x)=\left\{\begin{array}{l} -1 \text { if } x<4 \\ +1 \text { if } x \geq 4 \end{array}\right.\)
Therefore,f '(3) = −1
f '(5) = 1.

\(y={log \ x \over e^x }=e^{-x}.log \ x\)
\({dy\over dx}=e^{-x}({1\over x})+log \ x(e^{-x})(-1)\)
\(=e^{-x}[{1\over x}-log \ x]\).

\(y={cos \ x \over x^3}\)
\({dy\over dx}={x^3(-sin \ x)-cos \ x(3x^2)\over x^6}={-x^2(x \ sin \ x+3cos \ x)\over x^6}=-{(x \ sin \ x+3cos \ x)\over x^4}\)

\({dy\over dx}=xe^x({1\over x})+e^x.log \ x(1)+x \ log \ x(e^x)\)
\(=e^x+e^xlog \ x+xe^x log \ x=e^x(1+log \ x+xlog x).\)

\(y=x^2+{1\over x^2}-2=x^2+x^{-2}-2\)
\(\frac{d y}{d x}=2 x-2 x^{-2-1}=2 x-\frac{2}{x^3}\)

\(\frac{d y}{d x}=e^x+\cos x\)

\(y=\log _{10} x=\log _{10} e \log _{\ell} x\)
\(\frac{d y}{d x}=\log _{10} e \frac{d}{d x}\left(\log _e x\right)\left[\because \log _a b=\log _a c \cdot \log _c b\right]\)
\(=\log _{10} e\left(\frac{1}{x}\right)=\frac{\log _{10}{ }^e}{x}\) \(\left[\because d\left(\log _e^x\right)=d(\log x)=\frac{1}{x}\right]\)