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- (c) The equation of a curve is given by, x²y² + 4y = x (i) Show that dy_1– 2xy? 4 + 2x²y dx Find the gradient of the curve at (0,0). (iii) Hence, find the equation of the tangent to the curve at (0,0).Find the derivative of the function at Po in the direction of A. f(x.y,z) = xy + yz + zx, (1,- 1, - 2). A = 2i + 3j - 6k (DAf) (1. - 1. - 2) = (Simplify your answer.)5. Consider the function f(x, y) = sin(r² + y²). Find the derivative of f(r, y) at the point P(-/2,0, sin(2)) in the (a) direction of (4,3). Sketch the level curve that goes through the point P(-V2, 0, sin(2)). (b) On the same graph, include the gradient at point P. (c) point P (-V2,0, sin(2)). Find the direction in which the function f(r,y) has no change at the (d) What is the fastest rate of change possible at the pointP (-V2,0, sin(2))? (e) Find the equation of the line normal to the surface at the point P(-V2,0, sin(2)).
- A function z = (x, y) and a point P are given. Find the directional derivative of f in the indicated directions. f(x, y) = P = (1, 1) x²+y?+1' (a) In the direction of v = (1, - 1) (b) In the direction toward the point Q=(-2,-2)Find the directional derivative of f(x, y) = /4x + 3y at the point (2, 6) in the direction 57 radians. 34. Find the gradient Vf for the function f(x,y)= e* sin(7y) and the directional derivative at the point P (0,-) in the direction towards the point Q (3,-).
- Suppose that f (x, y) = 4e%* sin y. Find the directional derivative of f(x, y) in the direction of - 1, -4) and at the point (z, y) =(4, Find exact values. No decimalsQ10) use the Chain Rule to find the derivative of w- xy With respect to t along the path x-cost, y-sint.what is the derivatives value at t =Find the derivative of y with respect to x if y = (10x- 20x +20) ex. %3D dy dx (1,1) e Fntor voupncwor in the 2ncwor bay Show 4 (3x + 4) = 3 %3D - 3(3x + 4)² . 3 stmplify