\(\text { Given } f(x)=x+2\)
\(f^{\prime}(x) =1 \)
\(f^{\prime}(f(x)) =f^{\prime}(x+2)=1 \)

\( y=\log _{10} x=\frac{1}{\log _x 10}\)
Diff  w. r. to \(\log _x 10\)
\(y^{\prime}=\frac{-1}{\left(\log _x 10\right)^2}=-\left(\log _{10} x\right)^2 \)

\(x =a \sin \theta, y=b \cos \theta \)
\(\frac{d x}{d \theta} =a \cos \theta, \frac{d y}{d \theta}=-b \sin \theta\)
\(\frac{d y}{d x} =\frac{d y / d \theta}{d x / d \theta}=\frac{-b \sin \theta}{a \cos \theta}=\frac{-b}{a} \tan \theta \)
\(\frac{d^{2} y}{d x^{2}} =\frac{-b}{a} \sec ^{2} \theta \frac{d \theta}{d x}\)
\(=\frac{-b}{a} \sec ^{2} \theta \cdot \frac{1}{a \cos \theta} \)
\(= \frac{-b}{a^{2}} \sec ^{3} \theta \)

\(\frac{d x}{d t} =\frac{\left(1+t^{2}\right)(-2 t)-\left(1-t^{2}\right)(2 t)}{\left(1+t^{2}\right)^{2}} \)
\(=\frac{-2 t-2 t^{3}-2 t+2 t^{3}}{\left(1+t^{2}\right)^{2}}=\frac{-4 t}{\left(1+t^{2}\right)^{2}} \)
\(\frac{d y}{d t} =\frac{\left(1+t^{2}\right)(2)-2 t(2 t)}{\left(1+t^{2}\right)^{2}}=\frac{2+2 t^{2}-4 t^{2}}{\left(1+t^{2}\right)^{2}} \)
\(=\frac{2-2 t^{2}}{\left(1+t^{2}\right)^{2}}=\frac{2\left(1-t^{2}\right)}{\left(1+t^{2}\right)^{2}} \)
\(\frac{d y}{d x} =\frac{d y / d t}{d x / d t}=\frac{2^{\left(1-t^{2}\right)} /\left(1+t^{2}\right)^{2}}{-4 t /\left(1+t^{2}\right)^{2}} \)
\(=\frac{1-t^{2}}{-2 t}=-\frac{x}{y} \)

\(y =(a x-5) e^{3 x} \)
\(\frac{d y}{d x} =(a x-5) e^{3 x}(3)+e^{3 x}(a) \)
\(-13 =(-5)(3)+a \quad(\text { At } x=0) \)
\(-13 =-15+a \)
\(a =-13+15=2 \)
\(\therefore a =2 \)

\(\text { Given } e^{x+5 \log x}=e^{x} \cdot e^{5 \log x}=e^{x} \cdot e^{\log x^{5}}=x^{5} e^{x}\)
\(\frac{d}{d x}\left(e^{x+5 \log x}\right) =\frac{d}{d x}\left(x^{5} e^{x}\right)=e^{x} x^{5}+e^{x}\left(5 x^{4}\right) \)
\(=e^{x} x^{4}(x+5) \)

\(\text { Given } f(x)=x \tan ^{-1} x\)
\(f^{\prime}(x) =x \cdot \frac{1}{1+x^{2}}+\tan ^{-1} x \)
\(f^{\prime}(1) =\frac{1}{1+1^{2}}+\tan ^{-1}(1) \)
\(=\frac{1}{2}+\pi / 4 \)
\(\left(\because \tan 45^{\circ}=1\right)\)

\(y =m x+c \)
\(f(x) =m x+c \Rightarrow f(0)=c=1 \)
\(\therefore c =1 \)
\(f^{\prime}(x) =m \)
\(f^{\prime}(0) =1=m \)
\(\therefore m =1 \)
\(\therefore f(x) =x+1 \)
\(f(2) =2+1=3 \)

\(y=f\left(x^{2}+2\right) \)
\(\frac{d y}{d x} =f^{\prime}\left(x^{2}+2\right)(2 x) \)
\(\text { At } x =1, \frac{d y}{d x}=f^{\prime}(1+2)(2)=f^{\prime}(3)(2) \)
\(=5(2)=10 \)

\(\frac{d}{d x}\left(\frac{2}{\pi} \sin x\left(\frac{\pi^{\circ}}{180}\right)\right) =\frac{2}{\pi} \cos \frac{\pi x^{\circ}}{180^{\circ}} \frac{\pi}{180^{\circ}} \)
\(=\frac{1}{90} \cos x^{\circ} \)