 Hello and welcome to the session. My name is Mansi and I'm going to help you with the following question. The question says, find the derivative of the following function that is secant x. Now we find out the derivative of this function by using the first principle. According to the first principle f dash x is equal to limit h approaching to 0 function at x plus h minus function at x divided by h. So let us start with the solution to this question. Here fx is equal to secant x and secant x can be written as 1 by cos x. So function at x plus h will be 1 upon cos of x plus h. Therefore from the first principle we can say that f dash x will be equal to limit h approaching to 0 function at x plus h minus function at x divided by h. This we have already seen. So we simply put in the values now limit h approaching to 0 1 upon cos of x plus h minus 1 upon cos x the whole divided by h. Now this can be written further as limit h approaching to 0 the numerator we will have cos x into cos of x plus h and here we will have cos x minus cos of x plus h the whole divided by h. This is equal to limit h approaching to 0 to sin x plus x plus h by 2 into sin x plus h minus x by 2 divided by h into cos x into cos of x plus h. This happens because there is an identity that cos a minus cos b is equal to 2 sin a plus b by 2 into sin a minus b by 2. This is sin b minus a divided by 2. So we get this. Now this can be further written as here we see that plus x gets cancelled with minus x. So we have limit h approaching to 0 to sin h by 2 into sin 2x plus h by 2 divided by h into cos x into cos x plus h. Now we can take this separately and this separately we get limit h approaching to 0 to sin h by 2 divided by h into sin x plus h by 2 divided by cos x into cos of x plus h. Now multiplying the numerator and denominator by 1 by 2 we get limit h approaching to 0. Now 1 by 2 multiplied in the numerator this gets cancelled with 2. We have sin h by 2 divided by h by 2 into sin x plus h by 2 divided by cos x into cos of x plus h. This is equal to now we know that limit h approaching to 0 sin h by 2 by h by 2 is equal to 1. So we have 1 into sin x because here also we have applied limit h approaching to 0. So we are left with sin x divided by cos x into cos x because h is 0. So this is same as tan x into secant x. So our answer to this question is secant x tan x. I hope that you understood the question and enjoyed the session. Have a good day.