given that f, g, and h are differentiable functions and f(g(h(x))) = x, what is h'(x) in terms of f, f', g, g', and h?
I have on clue on how to even start it. please help . thank you.
To find h'(x) in terms of f, f', g, and g', you can use the chain rule for differentiation and some algebraic manipulation.
Let's start by differentiating both sides of the equation f(g(h(x))) = x with respect to x.
On the left-hand side, we have a composition of functions. By applying the chain rule, we can differentiate each function inside the composition one at a time, starting from the outermost function:
d/dx [f(g(h(x)))] = 1
Using the chain rule, we have:
f'(g(h(x))) * g'(h(x)) * h'(x) = 1
Now, let's solve for h'(x):
h'(x) = 1 / (f'(g(h(x))) * g'(h(x)))
Therefore, h'(x) in terms of f, f', g, and g' can be expressed as 1 divided by the product of f'(g(h(x))), and g'(h(x)).
Note that we don't need the specific forms of f, f', g, g', and h for this result. We primarily use the chain rule to differentiate the composition of functions and then solve for h'(x) in terms of the given functions and their derivatives.