# Differentiation of Composite and Implicit Functions

## The function whose values are found from two given functions by applying one function to an independent variable and then applying the second function to the result. If we have two functions u(x) and v(x), we can define a composite function w(x) = (u o v) (x) = u(v(x)). Let f be a real-valued function which is a composite of two functions g and h, i.e.,  f = g o h. Suppose, k = h(x) and if both dk/dx and dg/dk exist, then df/dx = (dg/dk (dk/dx). This is known as Chain rule. We can extend chain rule for three functions: Let f be a real-valued function which is a composition of three functions m, n and p, i.e., f = (m o n) o p. If t = p(x) and s= n(t), then df/dx = (dm/ds)(ds/dt) (dt/dx), provided all these derivatives exist. If f(x) = h(g(x)) then f’(x) = h’(g(x)) g’(x). In words we can say, differentiate the ‘outside’ function, and then multiply by the derivative of the ‘inside’ function. Keywords: Composite function, Chain rule, Implicit functions

To Access the full content, Please Purchase