Suppose f : R" → R™ is differentiable. Let B be an n by m matrix and define a new function g : R" → R" by g(x) = Bf(x). Show that g is differentiable and calculate its derivative.

Q: Consider f(t) = (cos t, sin t, t?) and g(x, y, z) = (x² + y² + z², 3x + 4y – 12z) and h = g(f(t)) =…

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Q: Find the linearization of x+1 f(x) = 2 – dt 1 + t“ at x = 1.

A: Given function is

Q: If U = lx+my,v=mx-ly then prove that partial derivative of x

A: Given: u=lx+my v=mx-ly   To show: ux=-vy and uy=vx

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

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Q: Find the first partial derivatives of the function. w = In(x + 2y + 7z) %3D aw Əx aw ду aw %3D az II…

A: Derivatives can be defined as rate of change of function with respect to an independent variable. It…

Q: Find all the first and second order partial derivatives of f(x, y) = - Vã+ In y. x3

A: While finding the partial derivative of a multivariable function, if we have to find the partial…

Q: 7). Given functions F(x, y) = (x²y x³ – y + 4, ) and G(u, v) = (u² + v, v) compute the Jacobian…

A: We have to compute the Jacobian derivative matrix of function G o F at the point (1.0).

Q: Find the first partial derivatives of the function. w = In(x + 2y + 5z) aw ax aw ду aw Əz II II

A: w= ln(x+2y+5z) To find: Partial derivative with respect to x,y,z.

Q: Find the Linearization L(x) of y = (1+x)¬1/² at a = 8. L(x) = %3D

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Q: Compute the partial derivative if z = A) - 3ry? B) 1 C) D) Does not exist.

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Q: If f (x, y) is a homogeneous function of degree n in x and y and has continuous first and second…

A: This question is based on Euler's theorem statement.

Q: Find all the second partial derivatives of the function a) f(x,y) = x* – 3x²y³. %3D | b) f(x,y) =…

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Q: Using the second fundamental theorem of calculus and chain rule, compute dz t dt. t+1 2x

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Q: 3. z = x+y x-y

A: Given:        z=x+yx-y Partial derivative with respect to x:      ∂z∂x=∂∂xx+yx-y      Treat y as…

Q: Find the first partial derivatives of the function. u = 8xY/z du %3D du %3D ay du %3D az

A: Please refer the attached image for complete solution.

Q: Find the partial derivative fx for the function f(x,y)=√16x+y³ |f₂(x, y) = O f.(x,y)= f₂(x, y) = 1…

A: We have to find derivative

Q: Show complete solution. Expand, that is, perform the indicated operations on the differential…

A: Given: xD(D+x)

Q: Let ƒ (x, y) = x³ − x² − 8x + 2xy − - - 12y². (a) Compute all first order and second order partial…

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Q: Recall the chain rule from differential calculus: f (g (æ)) = f' (9 (æ)) · g' (æ). d dx Find de tan…

A: We have to find

Q: If f is a function of x and y such that fxy and fyx are continuous, what is the relationship between…

A: Function: fx and fy are continuous function. Let us choose f(x,y)=x3y3+πxy Here fx=3x2y3+πy…

Q: Find the total derivative matrix Df at (x, y) = (1, 1) for the following function 2 y5 +2 a?y5 23 +…

A: Let's find.

Q: Find all second-order partial derivatives of the function f(x,y) = x + xy – 5x³ + In(x² + 1).

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Q: 11. Show that the Beta function, defined by B(x,y) = (1-t"-1 dt , for x> 0, y > 0, satisfies the…

A: the given function is βx,y=∫01tx-11-ty-1dt for x>0,y>0 substitute t=1-s therefore, dt=-ds when…

Q: Find all second-order partial derivatives of the given functions: (c) h(x,y) = In (x² + y)

A: Given; h(x,y)=ln(x2+y).

Q: - Find the total differential df of f (x, y) = (x² + y²) 3/2.

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Q: 2. Find the matrix of chain rule given u = xy + xz + yz; x = cos t, y = sin t, z = t.

A: Given

Q: Let D, be the linear transformation from C'(a, b] into C[a, b]. Find the preimage of the function.…

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Q: 2] write the chain rule formula for the derivatives Of and E , where Z=ZCrisit).

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Q: a.) If     f(x) dx   = 4x9 − 8x6 + C, find f(x). f(x) =

A: Given: ∫fxdx=4x9-8x6+c                                          ..1 for finding fx, we differentiate…

Q: Let Dx be the linear transformation from C'[a, b] into Cla, b]. Find the preimage of the function.…

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Q: Let Dx be the linear transformation from C'[a, b] into C[a, b]. Find the preimage of the function.…

A: Solution: Consider the given linear transformation : Dxf = 2x + 3 Since D is the linear…

Q: Express the autocorrelation function of the process {X'(t)} in terms of the autocorrelation function…

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Q: 1. Integrate the function f(x, y) = x – y over the triangle T with vertices (0,0), (1,0), and (1,2).

A: Given triangular region is bound by the vertices 0,0, 1,0 and 1,2. The sides of the given triangular…

Q: Write the matrix of partial derivatives of f(p, 0, ¢) = (p cos 0 sin ø, p cos 4)

A: Here in the given question, find the partial derivative of the given matrix:…

Q: Consider a regularizing term defined by where k!2* and the linear differential operator D is defined…

A: We are given the following regularizing term.                    ∫ℝN0DF(x)2 dx=∑k=0∞ak∫ℝN0DkF(x)2 dx…

Q: Find the linearization L(x) of y = e3" ln(x) at a = 1. %3D L(x) = ||

A: See the details solution in the details solution

Q: Find the linearization L(x) of the function at a. f(x) = cos(x), a = 3n/2 %3D

A: We have to find the linearization L(x) of the given function.

Q: д. L. = a(x, y)+b(x, y) +c(x, y). – d(x, y) -

A: Proof: Consider V be any vector space of all functions of two variable x  and y whose partial…

Q: Verify Young's Theorem f(x,y) = e(2x+y) In(xy)

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Q: Suppose that x2 = y2 = e and yz = zxy. Show that xy =yx.

A: Since, yz = zxy

Q: Find all the second order partial derivatives of the given function. f(x.y) = x+y

A: Partial differentiation

Q: Find all second partial derivatives of Z e

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Q: Find ý of the function x2 - e2x+3y - In(xy) = 0. %3D

A: Consider x2-e2x+3y-ln(xy3)=0We differentiate it implicity, we…

Q: Find the linearization of the function f(x)=x2+2cosx at a=0.

A: Linearization of a function f(x) at x = a is given by, Now,

Q: (b). Examine the behavior of e*+iy as x → -.

A: Euler's formula states that for any real number x  Any complex number z = x + iy,  can be written…

Q: Find all the second order partial derivatives of the given func f(x.y) = xy2 + yex + 5 A.

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Q: Find the first partial derivatives of the function. w = In(x + 6y + 9z) aw %3D ax aw ду aw Əz II II

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Q: Show that the functions In x , In(x2) linearly dependent (0, co) are on

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Q: Suppose that x2 = y2 = e and yz = zxy. Show that xy = yx.

A: Assume that x2 = y2 = e and yz = zxy.  Now from  yz = zxy we have  yz-1 = zxy-1z-1y-1 = y-1 x-1 z-1…