### Implicit differentiation

It sometimes happens that a function y is defined by a simple implicit relation F(x,y)=0 while its explicit expression y=f(x) can by very complicated, or y cannot be expressed with elementary functions of x. In this case, to get information about the the derivative of the function y,we can use implicit differentiation.

**For example** if we consider the curve (strophoid) defined by

**Question:** What is the slope of the tangent line for

**Remark:** All we need is

written y'(1), to answer the problem. Expressing y as a function of x is not necessary. We will solve the problem using implicit differentiation.

**Method: **Think that y is a mysterious function of x, written y(x). We need its derivative at x=1.
Let's try to solve the problem without finding an expression for y.

x(x²+y²(x)) is a function of x

2(X²-y(x)²) is a function of x too.

Those 2 functions are equal for any x therefore their derivatives are equal too. Let's differentiate both sides as functions of x. Of course, their derivatives equal

Congratulation! We just calculated an implicit differentiation!

**Answer:**The slope of the tangent line at x=1 is