Implicit differentiation

Implicit differentiation is differentiating by the help of chain rule on compositions of functions in which the inner functions are present implicitly.
For example, differentiating the expression `x^2 + y^2`, where y is a function of x, is implicit differentiation, but differentiating `x^2 + (x + 1)^2`, in which the chain rule is applied as well, is not implicit differentiation.

Implicit differentiation refers to the differentiation of compositions of functions in which the inner function is not directly stated. In the above example comparison of `x^2 + y^2 and x^2 + (x + 1)^2`, the former contains an implicit function ‘y’, while the latter contains an explicit function, ‘x + 1’, and so differentiating the former is known as ‘implicit differentiation’, while differentiating the latter is simply ‘differentiation’.

Implicit differentiation is different from the usual differentiation because the derivative of a function is different from the derivative of a variable. For example, if you are differentiating the equation `x^2 + y^2 = 1` with respect to x, the derivative of `x^2` will be 2x (from the power rule), but the derivative of `y^2` requires the use of chain rule because 'y' is a function of x.
Study the following example:
Derivative of x2 + y2 is
Using the power rule, the derivative of x2 is 2x:
Using the chain rule on y2,

  • The inner function is ‘y’ and the outer function is ‘2’ that is ‘whole squared’.
  • Substitute u = inner function, that is u = y, so you get u2. This may seem unnecessary but applying the chain rule correctly on it is essential for more complex ones.
  • Derivative of outer function is then
  • Substitute u = y again, so you get 2y.
  • Derivative of inner function is
  • Multiply the derivative of the inner function with that of the outer function, you get 2yy`

In the above discussion we have derived the expression x2 + y2 = 1 by the help of implicit differentiation.


  1. I am here to discuss about differentiation as-Differentiation is a method which is used to compute the rate at which a dependent output y changes with respect to the change in the independent input x and this rate is called as derivative of y with respect to x.

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