Your only error on the test was on implicit differentiation. For example if you have the expression x^3 + x^2 y^4 the derivative of x^3 with respect to x would be just plain 3 x^2. There is no y in this term so there is no dy/dx. The derivative of x^2 y^4 would be (x^2 y^4) ' = (x^2) ' y^4 + x^2 (y^4) ' , where the ' indicates derivative with respect to x. (x^2) ' doesn't include the y function so its derivative is just 2 x. However y^4 is a composite of f(z) = z^4 and g(x) = y(x) (recall that y is regarded as a function of x, so when we write y^4 we mean (y(x)) ^ 4), so its derivative is g ' (x) * f ' (g(x)) = y '(x) * 4 g(x)^3 = y ' (x) * 4 y(x)^3; we write this as 4 y^3 y ', and we understand that the 4 y^3 is just the derivative of the 'outer' function while the y ' is the derivative of the 'inner' function. Thus the derivative of x^2 y^4 is 2x y^4 + x^2 * (4 y^3 y '). You see that the y ' comes from the fact that y^4 is a composite function with respect to the variable x; there is no y ' on the derivative of x^2 because it is not a composite function--it's just a plain power function of x. You of course will write dy/dx instead of y '. Everything else you did with the implicit differentiation was correct; you just some extra dy/dx expressions.
Your only error on the test was on implicit differentiation. For example if you have the expression x^3 + x^2 y^4 the derivative of x^3 with respect to x would be just plain 3 x^2. There is no y in this term so there is no dy/dx. The derivative of x^2 y^4 would be (x^2 y^4) ' = (x^2) ' y^4 + x^2 (y^4) ' , where the ' indicates derivative with respect to x. (x^2) ' doesn't include the y function so its derivative is just 2 x. However y^4 is a composite of f(z) = z^4 and g(x) = y(x) (recall that y is regarded as a function of x, so when we write y^4 we mean (y(x)) ^ 4), so its derivative is g ' (x) * f ' (g(x)) = y '(x) * 4 g(x)^3 = y ' (x) * 4 y(x)^3; we write this as 4 y^3 y ', and we understand that the 4 y^3 is just the derivative of the 'outer' function while the y ' is the derivative of the 'inner' function. Thus the derivative of x^2 y^4 is 2x y^4 + x^2 * (4 y^3 y '). You see that the y ' comes from the fact that y^4 is a composite function with respect to the variable x; there is no y ' on the derivative of x^2 because it is not a composite function--it's just a plain power function of x. You of course will write dy/dx instead of y '. Everything else you did with the implicit differentiation was correct; you just some extra dy/dx expressions.