Question

How do you differentiate `f(x)=e^cosx`?

Answer 1

`f'(x) = -sinxe^cosx`

Explanation:

f 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 Let `y = f(x) = e^(cosx)`, Then:

`{ ("Let "u=cosx, => , (du)/dx=-sinx), ("Then "y=e^u, =>, dy/(du)=e^u ) :}`

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

`dy/dx = (e^u)(-sinx)`
`:. dy/dx = -sinxe^cosx`