Question 1: Consider the numbers 2 & 8. The average between these two quantities is 5, hence 2+8=10, 10/2=5. Now consider two arbitrary derivatives. It wouldn't make much sense to find the average between two unrelated derivatives, but suppose that f(x,y) was a function of both x & y. Now would it make sense to find the average between f'(x) & f'(y) to get f'(x,y)? I watched a YouTube video on finding derivatives with functions that contain more than one variable. The tutorial had you hold one variable constant to get the derivative of the other. It was a partial derivative. I'm just wondering why this is necessary & why you can't just be general about it & solve for f'(x,y)? I think I might know why. Suppose f(x,y)=x²+y³. You can't simply add the derivatives of those two variables together. That would be like adding two fractions together who's denominators are different. You can of course get around this by finding the lowest common denominator. I'm wondering if you can do the same, but with functions? Perhaps you could "stretch" x² or "contract" y³ so you can add them together? Question 2: In question 1, I mentioned that in the video I watched, I noticed that you couldn't solve for f'(x,y) & that you could only solve for the derivative of just one variable. Is this always the case? What if the derivatives of both the variables were the same? Suppose f(x,y)=x²+y². f'(x,y) is not 2x+2y? I see no reason why that wouldn't work.