Which of the following functions are linear transfonnalions?(a) L: R1 ....... R3 defined byL([II] IId) = [1I 1+ 1 11 2 11 1+112](b) L: Rz ..... R3 defined by L([1I1 lid) = [II I + l

Calculus 3 Week 4: Partial Derivatives ' Recall that the derivative f (x) represents the rate of change of the function as x changes. When we have partial derivatives, it just means we have more than one variable to keep track of. We will treat the other variables as constants while we differentiate the function with respect to the variable in question. Examples 2 a) f(x ,y)=3x+2y Let’s first differentiate with respect to . Remember we treat y as a constant. ∂x =3 ∂ f That’s it! Yes! Remember, constants have derivatives of zero! Now let’s differentiate with respect to y , treating x as constant. ∂y =4y ∂ f That’s not so bad right Ha! Get ready for so more difficult ones! b) f(x =x +sin xy +tan (z ) Now we have three variables to work with so we need to find 3 partial derivatives. First, let’s differentiate with respect to . ∂x 2 =3x +ycos(xy) ∂ f You did remember the chain rule right We have a function inside of a function so we need to differentiate the outside first and then differentiate the inside. The derivative of sin(xy) is cos(xy) and the derivative of xy with y as a constant is just y. Now, we differentiate with respect to y . There is only one term with y in it and we kinda found it already. ∂y =x