f(x) = 1/(x^2+1) and g(x) = sqrt (x), what is derivative of f(g(x))

Question

my questions is do I evaluate f(g(x)) and then solve for the derivative.

OR do I solve for the derivative using the f'(g(x)) * g'(x) rule?

Answer 1

You can do it either way. In this particular case it is probably easier to find the composite function and then find its derivative.

h(x) = f[g(x)] = 1/[(√x)²+1] = 1/(x+1) = (x+1)^-1

h'(x) = -1(x+1)^-2.

OR

f'(x) = -2x / (x²+1)²

f'[g(x)](g'(x)) = -2√x/ [ x+1]² (1/(2√x) = -1 / (x+1)²

Use this Desmos graph for validation:

desmos.com/calculator/krqm20guec