Question

Let f(x, y) = y cos (5x²) + 3x²644 Find fxy? (calculus 3)

Answer 1

Given the function
$f(x,y)= y\cos(5x^2)+3x^2e^{4y}$
And so, taking a partial derivative w.r.t. x, we get
$f_{x}(x,y)=\frac{\partial f(x,y)}{\partial x}=\frac{\partial }{\partial x} \left (y\cos(5x^2)+3x^2e^{4y} \right )$
  $\Rightarrow f_{x}(x,y)= -10xy\sin(5x^2)+6xe^{4y}$
And now, taking partial derivative w.r.t. y, we get

  $f_{xy}(x,y)=\frac{\partial f_x(x,y)}{\partial y}=\frac{\partial }{\partial y} \left (-10xy\sin(5x^2)+6xe^{4y} \right )$
  $\Rightarrow f_{xy}(x,y)=-10x\sin(5x^2)+24xe^{4y}$
Another way, you can first take partial derivative w.r.t. y and then take partial derivative w.r.t. x. But, you will get the same answer either way, because of the fact that $f_{xy}(x,y)=f_{yx}(x,y)$ .