Find dy/dx of the function 10x^2+4y^2=sqrt(7).
Is the answer 20x+8y=0
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.
To find the derivative of a function, you need to use implicit differentiation because this equation does not isolate y as a function of x.
Let's differentiate both sides of the equation with respect to x:
d/dx (10x^2 + 4y^2) = d/dx (sqrt(7))
Using the Chain Rule, the derivative of sqrt(7) with respect to x is 0 since sqrt(7) is a constant.
Now let's differentiate each term of the left side using the Power Rule for derivatives:
d/dx (10x^2) + d/dx (4y^2) = 0
Applying the Power Rule, you get:
20x + 8y * (dy/dx) = 0
Now, to solve for dy/dx, we need to isolate it:
8y * (dy/dx) = -20x
Divide both sides by 8y:
(dy/dx) = -20x / 8y
Simplifying further:
(dy/dx) = -5x / 2y
Therefore, the derivative of the function 10x^2 + 4y^2 = sqrt(7) with respect to x is dy/dx = -5x / 2y.
The answer you provided (20x + 8y = 0) is incorrect.