Question

Given that y = e^f(x), show that y' = f'(x)e^f(x).
Hence, find y' when f(x) = x.
Given that y = e^f(x), show that y' = f'(x)e^f(x).
Hence, find y' when f(x) = x.

Answer 1

this is just the chain rule. I assume you have seen the proof of that.

That said, just substitute x for f(x).

Answer 2

sorry sir pleasee show me how

Answer 3

There are lots of proofs of the chain rule on line. You can find one here, if you scroll down some

http://tutorial.math.lamar.edu/Classes/CalcI/DerivativeProofs.aspx

So, now you know that if

y = f(g) and g is a function of x, then
dy/dx = df/dg * dg/dx
You can see that treated as fractions, the "dg" factors cancel.

Now, how do you know that
d/dx = e^x ?
There are lots of proofs of that, as well, such as

http://math.stackexchange.com/questions/190773/proof-of-fracddxex-ex

(or, probably, in your own math text!)


So, if y = e^f(x)
y' = e^f f'(x)
If f = x, then f'=1, and we have the very useful result that

d/dx e^x = e^x