Mathematica Labs | Denis Shubleka

Subject: Calculus

Topic: Implicit Differentiation

Goal: Use Mathematica to implicitly differentiate equations.

Task 1

Suppose we are given an equation such as (x^3)+(y^3)=6xy  (Folium of Descartes),
and want to find or verify (dy/dx). We first move all terms to the left side:
(x^3)+(y^3)-6xy = 0  , and then differentiate after by writing y[x] instead of y, 
to indicate  that y depends on x.

lefthand=D[(x^3)+(((y[x]))^3)-6x*y[x],x]

Next, we solve for the unknown derivative:

Solve[lefthand==0,y'[x]]

Now, suppose we are working with parametric curves, where both x and y depend on
a third variable t (also known as the parameter). If we wanted to find (dx/dt) and (dy/dt), 
begin with:

lefthandside=D[(((x[t]))^3)+(((y[t]))^3)-6x[t]*y[t],t]

, and then solve for (dx/dt):

Solve[lefthandside==0,x'[t]]

, and finally, (dx/dt):

Solve[lefthandside==0,y'[t]]

Related Exercises/Notes:

ap-calc.github.io