Part 1/6:

An Introduction to Implicit Differentiation

Implicit differentiation is a powerful technique in calculus that allows us to find the derivative of a function even when it is not explicitly defined in the conventional sense. In contrast to explicit functions, where the variable y is expressed solely in terms of x (e.g., ( y = f(x) )), implicit functions can involve both variables in a more complex relationship, often making direct rearrangement difficult.

Understanding Implicit Functions

An implicit function is defined by an equation that cannot be easily reorganized to isolate y. For instance, take the equation ( x^2 + y^2 = 25 ) or ( x^3 + y^3 = 6xy ). These equations contain both x and y on the same side, making it challenging to rewrite y as a function of x.