Derivatives of Composite Functions

If f and g are the derivative functions in their domain, then f(g(x) is also differentiable. This is known as the chain rule of differentiation used for composite functions. (fog)'(x) = f'[(g(x)] g'(x). This also can be written as y = f(u) and t = g(x) , then dy/dx = dy/ dt . dt/dx

For example, consider y = tan²x. This is a composite function. By the chain rule, y = u² , where u = tan x

du/dt = 2u

dy/du = d/dx .tan x = sec ² x

dy/dx = dy/du . du/dx

= 2 u . sec ² x

= 2 tan x. sec ² x