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Let x = f(t) and y = g(t) be parametric forms with t as a parameter, then
$\frac{\text{dy}}{\text{dx}}=\frac{\text{dy}}{\text{dt}}\times\frac{\text{dt}}{\text{dx}}=\frac{\text{g}'(\text{t})}{\text{f}'(\text{t})},$ where $\text{f}'(\text{t})\neq0.$
On the basis of above information, answer the following questions.

1. The derivative of $\text{f}(\tan\text{x})\text{w.r.t.}\text{ g}(\sec\text{x})\text{ at}\text{ x}=\frac{\pi}{4},$ where f'(1) = 2 and $\text{g}'(\sqrt{2})=4,$ is:
   

1. $\frac{1}{\sqrt{2}}$
   
2. ${\sqrt{2}}$
   
3. 1
4. 0

2. The derivative of $\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)$ with respect to $\cos^{-1}\Big(\frac{1-\text{x}^2}{1+\text{x}^2}\Big)$ is:
   

1. -1
2. 1
3. 2
4. 4

3. The derivative of $\text{e}^{\text{x}^3}$ with respect to log x is:
   

1. $\text{e}^{\text{x}^3}$
   
2. $3\text{x}^22\text{e}^{\text{x}^3}$
   
3. $3\text{x}^3\text{e}^{\text{x}^3}$
   
4. $3\text{x}^2\text{e}^{\text{x}^3}+3\text{x}$
   

4. The derivative of $\cos^{-1}(2\text{x}^2-1)\text{w.r.t.}\cos^{-1}\text{x}$ is:
   

1. $2$
   
2. $\frac{-1}{2\sqrt{1-\text{x}^2}}$
   
3. $\frac{2}{\text{x}}$
   
4. $1-\text{x}^2$
   

5. If $\text{y}=\frac{1}{4}\mu^4$ and $\mu=\frac{2}{3}\text{x}^3+5,$ then $\frac{\text{dy}}{\text{dx}}=$
   

1. $\frac{2}{27}\text{x}^2(2\text{x}^3+15)^3$
   
2. $\frac{2}{7}\text{x}^2(2\text{x}^3+15)^3$
   
3. $\frac{2}{27}\text{x}(2\text{x}^3+5)^3$
   
4. $\frac{2}{7}(2\text{x}^3+15)^3$