What is the derivative of e^(sec x)?

1 Answer
Dec 9, 2016

d/dx e^secx = e^secxsecxtanx

Explanation:

If you are studying maths, then you should learn the Chain Rule for Differentiation, and practice how to use it:

If y=f(x) then f'(x)=dy/dx=dy/(du)(du)/dx

I was taught to remember that the differential can be treated like a fraction and that the "dx's" of a common variable will "cancel" (It is important to realise that dy/dx isn't a fraction but an operator that acts on a function, there is no such thing as "dx" or "dy" on its own!). The chain rule can also be expanded to further variables that "cancel", E.g.

dy/dx = dy/(dv)(dv)/(du)(du)/dx etc, or (dy/dx = dy/color(red)cancel(dv)color(red)cancel(dv)/color(blue)cancel(du)color(blue)cancel(du)/dx)

So with y = e^secx , Then:

{ ("Let "u=secx, => , (du)/dx=secxtanx), ("Then "y=e^u, =>, dy/(du)=e^u ) :}

Using dy/dx=(dy/(du))((du)/dx) we get:

\ \ \ \ \ dy/dx = (e^u)(secxtanx)
:. dy/dx = e^secxsecxtanx