How do you differentiate #y=e^(e^x)#?

Question

Emilia Aguilar · Accepted Answer

# dy/dx=e^(x+e^x) #

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 we have #y=e^(e^x)#, Then:

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

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

# dy/dx=(e^u)(e^x) # # :. dy/dx=(e^(e^x))(e^x) # # :. dy/dx=e^(x+e^x) #

William Abdalla · Answer

undefined To differentiate $ y = e^{e^x} $, you can use the chain rule. The chain rule states that if you have a function within a function, you differentiate the outer function first, leaving the inner function unchanged, then multiply by the derivative of the inner function.

So, for $ y = e^{e^x} $:

1. Differentiate the outer function $ e^{e^x} $ with respect to $ x $. The derivative of $ e^u $ with respect to $ u $ is $ e^u $, so in this case, the derivative of $ e^{e^x} $ with respect to $ e^x $ is $ e^{e^x} $.
  
2. Now, differentiate the inner function $ e^x $ with respect to $ x $, which is $ e^x $.

3. Multiply the results of steps 1 and 2 together to get the final derivative.

So, $ \frac{dy}{dx} = e^{e^x} \cdot e^x = e^{e^x + x} $.