Find dy/dx of the function 10x^2+4y^2=sqrt(7).

Question

Is the answer 20x+8y=0

Answer 1

No; that is not the right answer.

Differentiate both sides of the equation with respect to x. Because y is a function of x, this is called "implicit differentiation". You get
20 x + 8 y dy/dx = 0

Solve that for dy/dx

dy/dx = -20x/8y = -(5/2)(y/x)

Note that you did not have to solve for y(x) first to obtain this result for dy/dx. That is why implicit differentiation is so convenient.