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