Tangent line to a curve

[PredaNoob]

I was wondering if someone can please explain to me the logic they followed to solve this question? I've been trying to understand it, but can't.. They seem to be skipping things.

 

[biffboy]

Not sure what they are doing but to actually answer the question you can feed x=4, y=2 directly into their second line to get the required gradient.

 

[HallsofIvy]

The first sentence, beginning "The derivative of ...", is just using implicit differentiation.

The second sentence, beginning "Using the chain rule..." is nonsense because there is no definition or mention of the function "f". Are you sure you haven't acccidently copied that from a separate problem?

If you just replace x and y with 4 and 2 in the first equaton, you get 6y'+ (1/16)(16+ 4y')= 8+ 12y', a linear equation you can easily solve for y' at (4, 2) to complete the problem.