Question

When solving c^2 = a^2 + b^2 - 2ab(cos(θ)) for da/db using implicit differentiation, how does one know which a's to take the derivative of. In other words, the half way mark reads, 2aa'+ 2b - 2a(cos(θ)) - 2a'b(cos(θ)), why is the second a in the third term not a' when the others are?

Answer 1

think of 2abcos(θ) as a product rule.
Since θ is not dependent on b, just think of it as a constant.

Maybe it helps if you think of it in x and y terms With the product rule:
\((xy)' = x'y+xy'\)

So with your a and b's
\((ab)' = a'b + ab'\)
You can stick your constant 2cos(θ) in front of that product.
But b' disappears as it is just b^1

Answer 2

because only the a is implicit in the function, so keep the a', but the b is what you're actually differentiating with respect to