 Hi and how are you all today? The question says differentiate under root tan x with respect to x from first principle. Here let y be equal to under the root tan x. Right? Or we can say that y plus change in y is equal to under the root tan x plus change in x. Right? Further let this be the first equation and this be the second equation. Now on subtracting the second equation from the first we get change in y is equal to under the root tan x plus change in x minus under the root tan further we have now it adds. Now let us rationalize the numerator by multiplying and dividing it by under the root tan x plus change in x minus under the root tan x. Similarly this will be the term in the denominator also and on doing so we have it adds a minus b a plus b will give us a square minus b square that will be tan x plus change in x minus tan x upon under root tan x plus change in x plus under the root tan. Now further we can write it as sin x plus change in x upon cos x plus change in x minus similarly it can be written as tan x can be written as sin x upon cos and the numerator remains the same here we are not changing tan into sin and cos. Now further we have on taking the LCM we have sin x plus change in x into cos x minus sin x cos x plus change in x upon under root tan minus under root tan x. Change in y upon change in x is equal to limit change in x upon x and this all these part will be in bracket. Can we now written as on using the limits that is 1 upon cos x into cos x under the root tan x plus under root tan x. Can we written as secant square x upon under root tan require answer to this question. Hope you liked it have a nice day.