Given that f(x) has second derivative, f(0) = 0, f'(0) = 1, and the line integral | [xy(x + y) – f(x)]dx + [f'(x) + x²y] dy is path independent. Find thef (x) and calculate the integral - (x,y) | [xy(x + y) - f (x)] dx + [f'(x) + x²y] dy J(0,0)

Q: 1. Find the first partial derivative of z with respect to x if x'y+ y°z+ z'x = 11.

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Q: Find the second partial derivative for the function z = x7-7xy + 6y2 with respect to x.

A: Solution of given question is written in below steps.

Q: Find the first partial derivatives of the function (2x + 9y)'². = Z dz 1. дх dz 2. ду

A: To find the first partial derivatives of the function: z=(2x+9y)12 So, z is the function of x and y.

Q: Find the differential dy of the given function.

A: Given :              The given function is :                   y = 9 - x2 To find :            The…

Q: Find the differential dy of the given function.     y = 3x2/3

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Q: Find the differential dy of the given function. y = In√9-x² dy =

A: Given y=ln9-x2 Formula: ddx(ln u)=u'u where u is a function of x Now consider y=ln9-x2 The given…

Q: a²u(x, t) at² a²u(x,t) (-∞ 0 ) əx² и (х,0) — 4е-5)x] initial conditions ди -(х,0) %3D 0 at

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Q: Find a function w = ƒ(x, y) whose first partial derivatives are ∂w/∂x = 1 + ex cos y and ∂w/∂y = 2y…

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Q: Find the second order partial derivatives and of дхду aydx S(x.y) = 4x + In x + e² cos y + 9y + 3

A: d/dx(x^n)=nx^n-1 d/dx(cosx)=-sinx

Q: Find the second-order partial derivatives of the functions ƒ(x, y) = x + xy - 5x3 + ln (x2 + 1)

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Q: z = x siny + y sinx

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Q: If f(x, y, z) 2x5 + 3y5 - 8x2y2z2, what is the partial derivative of f with respect to z?

A: The given function is fx, y, z=2x5+3y5-8x2y2z2.

Q: Find the partial derivative of z with respect to x (32) for z = (3x² + 1) cos(x²y) O None of the…

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Q: Find the first partial derivatives with respect to x, y, and z. 7xz x+y

A: This question is related to differentiation, We will solve it using given information.

Q: Find the partial derivatives of the function f(x, y) = integral from x to y of cos(-6t^2 + 2t -…

A: Given: To Find: The partial derivative of f(x,y).   Fundamental Theorem of Calculus:

Q: Find the given partial derivative. f(x, y) = 2 sin(6x² – y). fy(0, ) =?

A: First we find the partial derivative with respect to y

Q: If f(x, y) = 3x² – 7y, then the total differential of ƒ at x = 1 and y = -2 is :

A: we need to find the total differential of the function  f(x,y) = 3x2 - 7y2 the total differential,…

Q: Find the differential dy of the given function. y = InV9 - x2 dy =

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Q: Find the solution of the differential equa­tion that satisfies the given boundary condition(s) .  f"…

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Q: Find the solution of the differential equa­tion that satisfies the given boundary condition(s) .…

A: Given equation is

Q: to Find the first partial derivatives of g(x,z) = x In (z dg dx ag dz

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Q: v = ) V In(2x-3y

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Q: (b) Find the general integral of the partial differential equation dz dz. = 7., X- which contains…

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Q: Find the first and second order partial derivatives for z x2 sin (3x + 2y).

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Q: Calculate the second-order partial derivative of f(x,y) =sin(x³y²) with respect to x and y, that is,…

A: given f ( x1 y) =sin ( x3 y2)   ∂2f∂x∂y=∂∂x∂f∂y

Q: Find the differential dy of the given function. y = In√√/9-x² dy =

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Q: (b) Solve the integral [(3x– y)dx+(2x+3y)dy along the curve x = y³ from (0,0) to (1, 1).

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Q: The partial differential equation = xy is classified as а. parabolic b. Hyperbolic/ Parabolic C.…

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Q: Compute the differential dz of the function. z = x*y" ,7,,11 dz

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Q: Find the differential dy of the given function. y = √√√√9-x²

A: Given y=9-x2 we need to find dy

Q: Find the solution of the differential formula in partial derivatives = x³ sent +1; y(x,t);…

A: Integrate twice with respect to t and then put conditions

Q: Find the Wronskian for the differential equatio: dy dx2 fip dx 3 10y = 0

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Q: Find the solution of the differential equa­tion that satisfies the given boundary condition(s) . y"…

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Q: Find the most general functions X(x) and Y(y), each of one variable, such that Z(x, y) = XY…

A: The given problem is to solve the given partial differential equation with given initial conditions…

Q: 5/(A)Evalute ex² dxdy. (B) If f(x, y) = x² ye(x+y²). Find the four seconed partial derivative.

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Q: Find the first and second order partial derivatives for z = x² sin (3x + 2y).

A: As we know that; The product rule: f·g'=f'·g+f·g' The chain rule: fgx'=f'gx·g'x And The first order…

Q: Find the partial derivative with respect to x of e^y-2*ln(x)-x^3y^2.

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Q: Find the number N such that the differential (y^3-2x)dx+Nxy^2dy is exact

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Q: Consider the nonhomogeneous differential equa 1 "(x) + y(x) = e '+ sin a

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Q: Find the differential dy of the given function. sec² x y x² + 2 sec 2x (2x²tan x + 4 tan x-2x)…

A: This question is based on quotient rule

Q: Find the differential of the function y = In(S - 2x) 2dx dy = (5-2x) dy = (5 – 2x )dx -2 (5-2x)dx…

A: Given : The function y = ln(s - 2x).   To find the differential of the function y = ln(s - 2x).

Q: Compute the differential dz of the function. 6.7 Z = x dz ||

A: Here, to find the differential dz of the function z=x6y7 Note that: dz=zxdx+zydy...(1)

Q: Find the solution of the differential equa­tion that satisfies the given boundary condition(s) . f"…

A: Given: f''-2f'+5f=0, f0=1  ,fπ4=0

Q: A more detailed version of Theorem 1 says that, if the f f(x, y) is continuous near the point (a,…

A: Given: (15). dydx=x-y (16).dydx=x-y

Q: Consider the following function: z = α x+ β y^2, where alpha and beta are constant parameters. Write…

A: To find the total differential dz and partial derivatives.

Q: Find the solution of the differential equa­tion that satisfies the given boundary condition(s) . h"…

A: Given: h''-4h'+5h=0 , h(0)=0 , h'(0)=-1 To determine: Solution of the differential equation.

Q: (xy(x² – y²) f(x,y) ={ x² +y² si x² + y² + 0 0 si (x, y) = (0,0)

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Q: Find the differential dy of the function y=(cos(x))(7x) a. dy=-7x(sin(7x))dx b. dy=(-cos(x))(7x)dx…

A: To Find the differential dy of the function y=(cos(x))(7x)

Q: What is the partial derivative dv/dy of the following function? v = 3x + 9xy -y/In(z) + cos (z + x)…

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