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

 

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