Partial Derivative Question, Totally Lost

Question

Question 1(a)Given the following functions: $$u(x,y) = yx^2-xy^2\\ v(x,y) = yx^3 + 2xy^5$$

I'm looking to evaluate the partial derivative of $u$ with respect to $x$ while $v$ is constant.

$$\left(\frac{\partial u}{\partial x}\right)_v$$

Answer 1

With $v$ being constant you know how the change in $x$ is related to the change in $y$. The chain rule you will use is

$$ \left(\frac{\partial u}{\partial x}\right)_v=\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}\frac{dy}{dx}$$ Since $v$ is constant, you have $$yx^3+2xy^5=c$$ So $$\frac{dy}{dx}x^3+3yx^2+2y^5+10xy^4\frac{dy}{dx}=0$$ From this equation solve for $\frac{dy}{dx}$ and put into the chain rule's partial derivative equation for $u$.