Implicit differentiation of this equation

I am learning Differentiation in Matlab I need help in finding implicit derivatives of this equations find dy/dx when x^2+x*y+y^2=100 Thank you.

[madhan ravi]

Is y a function of x and x(t)?

Answer 1

[Walter Roberson]

syms x y
diff(solve(x^2+x*y+y^2==100,y),x)

You will get two solutions because there are two distinct y for each x

If you want, you could continue

syms dy; solve(dy == diff(solve(x^2+x*y+y^2==100,y),x),x)

to get x in terms of dy

[Ankit Gupta]

what if i need answer in terms of x and y?

[Walter Roberson]

I do not know how to solve it in terms of x and y.

Answer 2

[Mbar]

Consider implicit function . It is not always possible to solve analytically for . However, almost always you can use the Implicit Function Theorem:

, as long as .

( are partial derivatives: ).

Thus, define the implicit function, , and the derivative is

. In Matlab (using Symbolic Math Toolbox):

syms x y %Declaring symbilic variables
F(x,y) = x^2 + x*y + y^2 - 100 %Declaring implicit function

% Using Implicit Function Theorem
dy_dx = - diff(F,x)/diff(F,y)

% Answer:
% -(2*x + y)/(x + 2*y)

This derivative is a function of both x and y. However it has a meaning only for pairs which satisfy the implicit function . You can solve for such points using what Walter Roberson suggested. For example, solve for y as a function of x, and substitute :

double(subs(solve(F, y), x, 10))

This gives two points which satisfy the implicit function: , and . You can calculate the derivatve at these points for example:

x0 = 10; y0 = -10;
F(x0, y0) %answer 0, i.e. the implicit function is satisfied.
dy_dx(x0, y0) %answer 1

x1 = 10; y1 = 0;
F(x1, y1) %answer 0, i.e. the implicit function is satisfied.
dy_dx(x1, y1) %answer -2

[UMAIR]

If we want to find d^2y/dx^2 ?