Is this differentiation correct?

Question

$J(x,y')=\int_1^2 xy'(x)+(y'(x))^2dx = \int_1^2{f(y,y^\prime,x)}$

Need to find $\frac{d}{dx}(\frac{\partial f}{\partial y^\prime})$

$\frac{\partial f}{\partial y^\prime}=x+2y'(x)$

$\frac{d}{dx}(\frac{\partial f}{\partial y^\prime})=1$

Is this correct, or what is the right solution?

Answer 1

When you do ${d\over dx}(x + 2y^\prime)$, you need to remember that $y^\prime$ is a function of $x$, so the result is $1 + 2y^{\prime\prime}(x)$.