Use the general chain rule to find DH (1,-1) given G(U)=G(u, v) = and H(U)= F(G(U)). /u²+3v² +2 F(X) = F(x, y, z) = [ - x² + y² + 2z² x² - y² '],

Q: What are the P(x) and Q(x) of y' =y-xy³e2Xrespectively?

A: Given the differential equation is: Since

Q: Find y as a function of a if y(0)2, y'(0) = 3, "(0) = -13. y(x) = y″ − 4y″ − y′ + 4y = 0,

A: Given a differential equation:initial conditions are:

Q: 5c) Find the partial derivatives?     y = 2x2+3xz+85w-8z2

A:

Q: Find fx, fy, fz. fx = + +2² f(x,y,z) = e¯ (4x²+2y² 12=7 Z

A: f(x,y,z)=e-(4x2+2y2+z2)we know (i)  ∂ ∂x(xn)=nxn-1(ii)  ∂ ∂x(constant)=0

Q: Calculate r' (t) and T(t), where r(t) = (2t - 7,2 – 5t, 3 + 5t). r' (t) = (2 T(t) = -5 5

A:

Q: Consider the function g(x, y) = x² + xy + y² (1 + 1/2t, 2 + V3/2t), use the (a) Given the line l(t)…

A: Given,              g(x, y)=x2+xy+y2

Q: Let f(x, y) = x² + 5y². (a) Find f (5, 1) and f,(5, 1). fx(5, 1) fy(5, 1) =

A: Find

Q: Let F(x, y, z) = x3³yz² - 6x²yz + 9xz - 2y³z. Find F Fxyz' 3x²yz² - 12xyz +9z+ x³z² − 6x²z − 6y²z +…

A: This question is based on Partial Derivative.

Q: x²y" + 3ry – 24y = 0, y(1) = -8, y'(1) = –7. y =

A:

Q: let h(y)=(y-3y2)5(y2+2)6  h'(y) in simplified form.

A:

Q: u = Vx + 5 (x + 1) then du/dx is Зх — 11 Зх + 11 2 Vx + 5 2 Vx + 5 A 3x + 10 Зх — 10 2 Vx + 5 2 Vx +…

A:

Q: Find the first order partial derivatives of: (a) g(x1, x2) = (3x1 + 2x2) 5. Ans: f1=? f2=? (b)…

A:

Q: Find r'(t), r"(t), r'(t) · r"(t), and r'(t) × r"(t). P(C) = 21²1 - 48 + ²P²H 2³k r(t) 3 (a) r(t) 3it…

A:

Q: Utt u(x, 0) ut (x, 0) = = = 4uxx x² + 3x

A: USIND D'ALEMBERT'S SOLUTION FOR ONE DIMENSIONAL HOMOGENOUS WAVE EQUATION ,WE HAVE

Q: Give an example of ANY one nonlinear (i.e. powers of r, y can't just be 1) function z f(r, y) such…

A:

Q: Solve Fyy + 3 Fxy + 2 Fxx = 2x, F(x, 0) = -x, 2, F(x, 0) = 0. F(x, y) =

A: General solution is given by complementary factor + particular integral.

Q: Find y as a function of a if y(1)9, y'(1) = -5. y = x²y" - 15xy' + 64y = x²,

A:

Q: f(x, y) = V 2 2 3x² + 4y²

A:

Q: For f(x, y, z) = x² - y² + z², find f(2, -1, 4).

A:

Q: Find y as a function of if y(0) = 19, y'(0) = 19, y"(0) = 4, y"(0) = 0. y(x) = y(4) 4y" + 4y = 0,

A: y(4)−4y′′′+4y′′=0y(0)=19,y′(0)=19,y′′(0)=4,y′′′(0)=0

Q: I. Find y', y" and y"" of the following functions. a. y = 2x-x'+ 5 b. y = (r²+2) x-1 II. Use the…

A: I. (a).y=2x4-x3+5diff w.r.t xdydx=2(4x3)-3x2+0y'=8x3-3x2diff w.r.t x…

Q: Find y as a function of a if y(0) = -2, (0) = 9, '(0) = -62. y(x) = y" - 4y" -y + 4y = 0,

A: Given that y''' -4y'' - y' + 4y = 0 ,   y(0) = - 2 y(0) = 9 '(0) = -62.

Q: First write the function y = Vx2 +1 as a composition of y = f(u) and u = g(x), and then dy apply…

A: y=x2+1y=f(u), u=g(x)To find:dydx

Q: For the function z = - 8x³ - 2y² - 2xy, find a z(1,-2), andz(1,-2). dz dz a ax' dy ax

A:

Q: Consider f(x, y) = xy(²-x²) x² + y² 0 if (x, y) = (0,0) if (x, y) = (0,0) . Show that fay(0,0) = 1…

A:

Q: The graph of y=x'-3x-24x +32 is concave upward on the interval (1, oo) because: O'=y"=0 Oy' is…

A: The given function is y=x3-3x2-24x+32

Q: Find y as a function of a if y" - 3y" y + 3y = 0, y(0) = -7, y' (0) = -4, y" (0) = 9. y(x) =

A:

Q: fx(2, 2) = fy(2, 2) = # f(x, y) = x² + 4x¹y.

A:

Q: b) Use the chain rule for partial derivative to find 2 when z = f(x, y) = x²y + ln(x + y) and (x, y)…

A: