Define a function f : R² → R² by f 2.у 1. Show that f is differentiable on A = E R² |x > 0, y > 0} and compute the derivative.

Q: Let f'(a) denote the derivative of the function y = f (x), calculated at x a. Prove that the…

A: Given,                  y= f(x)                      (1)

Q: find the derivative of the function at P0 in thedirection of v. ƒ(x, y, z) = x2 + 2y2 - 3z2, P0(1,…

A: f(x,y,z) = x2 + 2y2 -3z2  v = i + j + k  P0(1, 1, 1)

Q: Let y = (x + sin(x))". Find g(1) and f() so that y = (/o g)(x), and compute the derivative using the…

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Q: (a) Find the directional derivative of f (x,y) = In(xy) at (-) in the direction of (1,1).

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Find the derivative of the function at Po in the direction of u. f(x, y, z) = In(x 2 + 3y 2 + 9z 2),…

A: Since you have posted multiple questions i can do first question as per our company guidelines…

Q: Let y be a function of x and suppose *= In(y<-x²). What is the correct result for the derivative of…

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Q: Find the derivative of f(z)= In(z-+-3z+2i). O (A) ƒ'(2) = O (B) ƒ'(2) = O (C) ƒ'(2) =1 – O (D) f'(z)…

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Q: Use the linear approximation of f(x) = at r = 0 to approximate f(-0.1). 1+r

A: Use the linear approximation of f(x)=1/(1+x) at x=0 to approximate f(-0.1).

Q: Differentiate. f(x) = In ( In (8x)) f'(x) = Compute the following derivative using the method of…

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Q: Let f(x) = e2sinz 1. Use the local linear approximation of f at x = 0 to approximate e2 sin(0.01

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Q: Use the second derivative f"(x) = x(x+ 1)(x-3) to determine where the function f(x) is concave down.…

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Q: Find the direction at which the function f(x, y, z) = x In(e – az) increases fastest at (-3,0, 0) O…

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Q: Find the derivative of the function. g(x) = [" (Hint: " u2 - 1 f(u) du + 5x du f(u) du = f(u) du J5x…

A: Fundamental theorem of Calculus: If f is a continuous function on a,b such that, F(x)=∫axftdt, then…

Q: Find the derivative of h(z)=(az+b)/q  . Assume that a, b and q are constants. h′(z)=

A: Given function is, h(z)=(az+b)q where a, b &amp; q are constants.

Q: Let f(2) = x* – i(y + 1)°, (z = x + iy), (a) Determine the points where f is C-differentiable, and…

A: Given :-  let f z = x4 - i  y + 13 where z = x + iy (a) The points where f is ℂ - differentiable ,…

Q: Use the Fundamental Theorem of Calculus to find the derivative of y- " cos t dt t12

A: Given that  y=∫-9xcos tt12dt

Q: · F(x) = 5* (S₁²¹6 (√₁+ u²) du) dt

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Q: Let f(x) = sin? (x). Define F(x) | f(t) dt. What is F" -) ?

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Q: Suppose F(z) = | e dt. Use the Fundamental Theorem of Calculus to evaluate the derivative. 13 F'(포)…

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Q: Let y be a function of x and suppose *= In(x--y'). What is the correct result for the derivative of…

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Q: Consider the function f (x, y) = x²y³ – 3xy³. Compute the value of the derivative frvy at the point…

A: Given function is f(x,y) =  x2y3 - 3xy3 we have to find fxyy at the point (2,1).

Q: tan-lt t +0 t f(t) = 1 t = 0.

A: Given that, f(t)=1 when t=0. Differentiate the function with respect tot t implies, f'(t)=0  when…

Q: Let f be the function with derivative given by f' (z) = sin z +I cOs z for 0 <z < 7. On which of the…

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Q: Let f (z) = (11z+ 10) (i – z). What is the real part of the derivative f' (1 + i)?

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Q: d rcos t sin(x²) dx dt

A: Given,            ddt∫1cos t sinx2 dx

Q: Find the derivative of the function at Po in the direction of u. f(x, y) = In(-10x - 7y), P o(-4,…

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Q: Let z = x+iy and f(2)=√√|xy|. Show that ƒ(z) satisfies the Cauchy-Riemann equations at the origin,…

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Q: Let f(1,y) = (r+ay°,r²y°,2r + by), g(u, v,w) = (3u² + bv² + w°, uvw). Find the derivative of…

A: Let f(x, y)=x3+ay2, x2y3, 2x+by,  gu, v, w=3u2+bv2+w3, uvw. We need to find the derivative of fgu,…

Q: Let f (z) = (5z +3) (i – xỳhat is the real part of the derivative f' (1+ i}

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Q: Let u(t) = (3,3t,t) and v(t) = (3t, - 2t,2). Compute the derivative of the following function. u(t)…

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Q: 10. Let f(x, y) = x² + y³. Calculate the directional derivative of f in the direction of = (4, 3) at…

A: The given function is: f(x,y)=x2+y3Direction vector is:  v→=4, 3Given point:  P=(1, 2)

Q: S:R R, f(x) =a", (a > 0, + 1)

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Q: Compute the derivative of h(sin(x)) at x = T/6, assuming h'(0.5) = 10:

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Q: Let f(2, y) = In(' + 2 xy). Find the directional derivative of f at the point (1,1) in the direction…

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Q: Find the derivative of the following function (2t – 1)* dt F(x) = using the Fundamental Theorem of…

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Q: Suppose that F(x) = /. f(1) dt, where V6 +u du. f(t) =

A: Given F(x)=∫1x f(t) dt where f(t)=∫1t4 6+u5u du

Q: Suppose that f(a) = f(b) = 0 and the second derivatives of f exist on the closed interval [a, b].…

A: Prove that ∫abx-ax-bf"xdx=2∫abfxdx.

Q: Show that w=f(z) =3ix+y dose not satisfy C.R.E. thus it is not analytic function

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Q: The fundamental theorem of calculus says that the integral of the derivative of a function is the L…

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Q: Find the derivative of the function at Po airection oT u. f(x, y) = 3x 2 + 4y, P o(8, 7), u = 3 i -…

A: 1. Given f(x, y) = 3x2 + 4y and u = 3i - 4j u^ = 1|u|u = 15(3i - 4j) ∇f·u^ = (i^ddx(3x2+4y) +…

Q: Use the limit to find partial derivative of…

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Q: Find the directional derivative of f at the given point in the direction indicated by the angl f(x,…

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Q: Suppose F(z) = e dt. Use the Fundamental Theorem of Calculus to evaluate the derivative. F'(z) = dt…

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Q: compute the derivative of f ◦ g. f (u) = sin u, g(x) = 2x + 1

A: Let By chain rule of differentiation, we have

Q: Use the First Part of the Fundamental Theorem of Calculus to find the derivative of the following…

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Q: Suppose that f:I → R and g: I → R are differentiable at c E I. Then, a) If k E R, then the function…

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Q: Let f(x, y) = (x² +(G))°. Determine the directional derivative of f at the point (1 0) in the…

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Q: Suppose w is a function of x, y, and z, which are each functions of t. Explain how to find dw/dt.

A: Given data:"w" is a function of (x,y,z). Therefore, w(x,y,z). And the function x, y and z are the…