Given the function fxy x3y3 exy i compute the gradient...

Given the function f(x,y) = x^3 + y^3 - e^(xy) i.) Compute the gradient vector of f. ii.) Compute the directional derivative of f at the point (0,1) in the direction of the vector v = <2,1>. Remember to normalize v. iii.) Determine the maximal rate of change of f at the point (0,1). iv.) Determine the direction in which the maximal rate of change of f at the point (0,1) occurs.

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00:01 Hi, in this question we are given with the function xy that is equals to x cube plus y cube minus e raise to par xy.

00:11 For the first part we need to find the gradient of the function.

00:16 As we know, gradient of function can be calculated as del of f and this would be equals to curly by curly x times x cube plus y cube minus e raise to par xy xxx.

00:30 Plus curly by curly y x cube plus y cube minus e raised 2 par xy j cap and solving for the partial derivative we get del f is equals to 3x square minus y times e raised 2 par xy i cap plus 3y square minus x times e to par x y cap now solving for the directional derivative second part as we know the directional derivative can be calculated as equal to gradient of function with the dot product of unit direction vector and as we have already calculated the gradient function now we need to find the gradient function at the given point of 0 and 1 where we are having x as 0 and y as 1 so substituting this value we get 3 times 0 square minus 1 times e raised 2 0 i cap plus 3 times 1 square minus 0 j cap or solving this we get this to be equals to minus i cap plus 3 j cap and solving for the unit direction vector and this would be equals to the given factor that is 2 i cap plus j cap divide by its magnitude 2 square plus 1 square and solving this we get this to be equals to 2 i cap plus j cap divided by under root 5 now we can write the directional derivative that is equals to gradient function minus i cap plus 3 j cap dot product with 2 upon under root 5 i cap plus 1 upon under root 5 j cap and solving this further we get this to be equals to minus 2 upon root 5 plus 3 upon root 5 and solving this we get our final result for the directional derivative and this would be equals to 1 by under root 5 now moving on to the next part third we need to find the maximum rate of change of the function...

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