If f and g are functions such that f(g(x))=g(f(x))=x for all x in their domains, and if f(a)=b and f'(a)=c, then g'(b) =?

a) g'(a)=1/c
b) g'(b)=1/c
c) g'(b)=-1/c
d) g'(b)=1/a
e) g'(a)=-1/c

To find the value of g'(b), we can use the chain rule of differentiation. The chain rule states that if we have a composition of two functions, such as f(g(x)), then the derivative of this composition can be found by multiplying the derivative of the outer function by the derivative of the inner function.

In this case, we have f(g(x)) = x. Let's differentiate both sides of this equation with respect to x.

d/dx [ f(g(x)) ] = d/dx [ x ]

Using the chain rule, the left-hand side becomes:

f'(g(x)) * g'(x) = 1

Now, let's substitute the values given in the question. We know that f(a) = b and f'(a) = c. So, f'(g(x)) can be written as f'(b). Similarly, g'(x) can be written as g'(a).

Plugging in these values, we have:

f'(b) * g'(a) = 1

Now, we want to find g'(b), so we need to rearrange the equation:

g'(a) = 1 / f'(b)

Finally, substitute the given value f'(a) = c:

g'(a) = 1 / c

Therefore, the correct answer is option a) g'(a) = 1 / c.