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A second approach follows the second method in part (b) above.
∂z
∂z
dz = ∂ x dx + ∂ y dy where the partial derivatives were calculated
∂z ∂z
ferentiation. Then start rewriting this as dz = ∂ x , ∂ y · dx, dy
4, −5 · dx, dy = 4, −5 dx, dy cos(θ) and ﬁnally continue
late the direction of maximum increase. First observe that
using implicit dif= ∇z · dx, dy =
as before to calcu- These two approaches say exactly the same thing, but they use slight diﬀerent notation.
Both notations are widely used so it’s important to get used to translating between
the two.
(d) Find the directional derivative of z = f (x, y ) at P (2, 1, −1) in the direction ...

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