Hi there!
We can use implicit differentiation with respect to x:
[tex]3xy = 4x + y^2[/tex]
If a term with 'y' is differentiated, a 'dy/dx' must be included.
We can differentiate each term separately for the explanation.
3xy
We must use the power rule since 'x' and 'y' are both in this term.
Power rule:
[tex]f(x) * g(x) = f'(x)g(x) + g'(x)f(x)[/tex]
[tex]3xy \\\\f(x) = 3x\\g(x) = y \\\\f'(x)g(x) + g'(x)f(x) = 3y + 3x\frac{dy}{dx}[/tex]
Now, we can do the others.
4x
This is a normal power rule derivative.
[tex]f(x) = 4x\\f'(x) = 4[/tex]
y²
Since we are not differentiating with respect to y, we must include 'dy/dx'.
[tex]f(x) = y^2\\f'(x) = 2y\frac{dy}{dx}[/tex]
Combine the above:
[tex]3y + 3x\frac{dy}{dx} = 4 + 2y\frac{dy}{dx}[/tex]
Rearrange to solve for dy/dx.
Move dy/dx to one side:
[tex]3x\frac{dy}{dx} - 2y\frac{dy}{dx} = 4 + 3y \\\\[/tex]
Factor out dy/dx and divide:
[tex]\frac{dy}{dx}(3x- 2y) = 4 + 3y \\\\\boxed{\frac{dy}{dx} = \frac{4+3y}{3x-2y}}[/tex]
