How do you differentiate f(x)=e^(tansqrtx) using the chain rule.?

1 Answer
Apr 12, 2016

f'(x) = e^tan(sqrtx) * sec^2sqrtx/(2sqrtx)

Explanation:

Chain rule is basically a lot of substitution.

let u=tan(sqrtx)

now f(x) = e^u

so if we try to take the derivative now we apply chain rule to get:
f'(x) = e^u * (du)/dx

so now we are on a mission to find (du)/dx...

We are going to need to rewrite our u equation to something like this:
u=tan(w)
w=sqrtx

Derive u

(du)/dx = sec^2w * (dw)/dx

Now to find (dw)/dx...

w=sqrtx
Yay we know how to do this one!

(dw)/dx= 1/2x^(-1/2)=1/(2sqrtx)

now substitute back into (du)/dx

(du)/dx = sec^2w * 1/(2sqrtx) = sec^2w/(2sqrtx)

and finally substitute back into f'(x)

f'(x) = e^u * sec^2w/(2sqrtx)

now change everything back to terms of x (you could have done this step earlier)

f'(x) = e^tan(sqrtx) * sec^2sqrtx/(2sqrtx)