Derivative of Implicit Functions

Till now, the functions that we have discussed, are explicitly functions of x. We have defined y in terms of x. Suppose we have an equation f(x,y) = 0, which cannot be put in the form of y=f(x) to differentiate in the usual way, we can still differentiate the equation f(x,y) = 0. This function in which y cannot be expressed in terms of x, is called an implicit function.Till now, the functions that we have discussed, are explicitly functions of x. We have defined y in terms of x. Suppose we have an equation f(x,y) = 0, which cannot be put in the form of y=f(x) to differentiate in the usual way, we can still differentiate the equation f(x,y) = 0. This function in which y cannot be expressed in terms of x, is called an implicit function.

x³+ y³- 9xy = 0 is an example of an implicit function.

The process of differentiating implicit function is called implicit differentiation.

Rules for Implicit Differentiation

Step 1:

Step 2:

Differentiate the terms containing x, y or both xy with respect to x.

While differentiating the terms containing y or power of y, first

Step 3:

Step 4:

Example:

Differentiate the following implicit equation

ax² + 2hxy + by² + 2gx + 2fy + c = 0

Step 1:

ax² + 2hxy + by² + 2gx + 2fy + c = 0

Step 2:

Differentiating with respect to x, we have

Step 3:

Step 4: