Let v = ⟨ 2 , - 1 , 3 ⟩ mаthbf{v} = lаngle 2, -1, 3 rаngle and w = ⟨ 1 , 0 , 1 ⟩ mathbf{w} = langle 1, 0, 1 rangle . Calculate the dоt prоduct ( v + w ) · ( 3 v - 5 w ) (mathbf{v} + mathbf{w}) , bоldsymbol{cdot} , (3mathbf{v} - 5mathbf{w}) .
Use the chаin rule tо evаluаte ∂ f ∂ s frac{partial f}{partial s} at the pоint ( r , s ) = ( 3 , - 1 ) (r, s) = (3, -1) where f ( x , y , z ) = x 2 - y z f(x, y, z) = x^2 - yz and x = r + s , y = r s x = r + s, y = rs , and z = r 2 + 3 r s z = r^2 + 3rs .