af AMPLE 3 If f(x, y) = sin). calculate and 5 + y ax LUTION Using the Chain Rule for functions of one variabl 3x 5 +y af a(_3x ax\5 + y ax ])) af a 3x ay5 + y. ду

Q: Let f(x, y, z) = In(x² + 2) + y²e== – cos(2yz). Find fzyz- %3D

A: Given that: f(x, y, z)=ln(x2+z)+y2exz-cos(2yz)

Q: (c(2z+y), %3D ue of c in the joi F of W.

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Q: If f(x, y, z) = excosy + z, show that fxx + fyy + fzz = 0

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Q: Suppose that z = x/y, where x and y are differentiable functions of u and v such that x = dx/dv 3, у…

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Q: Determine if the function satisfy the Cauchy-Riemann equations. Show your soluition. 6. f(z) = z?

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Q: Suppose that z = f(x, y) is differentiable at the point (1024, 64) withf, (1024, 64) = 4 and f,…

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Q: Prove that: cscz- cot z = 1 2. 2. %3D

A: We need to prove the identity  csc2z-cot2z=1

Q: Given : 6 (x, y, z) Compute the gradient 3 6y³z³ - 5 ex at = PA (2,2,1.5) of G (x, y, z)

A: Given  G(x,y,z)=6y2z3-5e-x We have to find the gradient at the point PA(2,2,1.5)

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

A: Given: Dx be the linear transformation from C'a,b into Ca,b. To find: Preimage of the function…

Q: 2. If F(x, y, z) = 2z3 – 3(x² + y²)z, then verify that Frx + Fyy + F2 = 0. %3D

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Q: Suppose that z = f(x, y) is differentiable at the point (1024, 256) withf(1024, 256) = 2 and f,…

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Q: Suppose that a duck is swimr - perature is given by the formı v²sin x. Find dT/dt in two chain rule…

A: Given query is to find the derivative of function T.

Q: Assume that z = ln (ƒ(w)), w = g(x, y), x = sqrt(r - s), and y = r^2s. If gx(2, -9) = -1, gy(2, -9)…

A: Given Now, we will find partial derivative of w with respect to r and s by using chain rule

Q: Choose the gradient of the function f defined by f(x, y, z)= In[1 + (17xy + xz +9yz)²].…

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Q: Suppose that X P(X < 3Y). unif(-2, 2) and Y ~ unif(-1,2) are independent continuous RVs. Find P (|X…

A: Given: X ~ U (-2, 2) Y ~ U (-1, 2) X and Y are independent.

Q: Determine the local linear approx imation of flx,4,z) =(1+x) (l-y³) (l-2)²' at the point CT, 2,3)

A: The local linear approximation: Lx, y, z of the function f(x, y, z) at the point x0, y0, z0 is…

Q: Use the chain rule to eX Pressdy intermsof X and Y: and t= at X=2

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Q: aーメ ターメ2.y to い-X to cylindHcal coordin ates

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Q: Show that f(x,y)=exyx is differentiable at point (1,0) using the definition of differentiability.

A: A function is called differentiable, if the derivative of function exists at each and every point in…

Q: Find curl F if F(x,y,z) = 16yºzº i - 8x³z8j - 7xy4 | curl F = (-28xy3 + 64x³z7) i+ (7y4 + 80y5zª)j+…

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Q: Compute the gradient of the function at the given point. f (x, y, z) = In(x² + 7y – 3z²) ; (7,7,7) O…

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Q: Suppose that f(æ, y) f(x, y) in the directional (1, – 1) and at the point = xy. The directional…

A: Here, the given function is: f(x,y)=xy So, ∇f=∂f∂x,∂f∂y⇒∇f=y,x⇒∇f(4,-3)=-3,4

Q: Suppose that z = f(a, y) is differentiable at the point (1024, 1024) with, (1024, 1024) = 5…

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Q: Suppose that f(r, y) the point (r, y) 3 (6, ry. The directional.derivative of f(z, y) in the…

A: Given:  fx, y=xy To Find:  To find the direction derivative of fx, y at the point x, y=6, -4, in the…

Q: Let F = (2yz – 3x) i + (3xz + 3y) j – (2xy+ ∞z) k. Find div F and curl F.. |

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Q: Show that the function f defined on [0, 1] by the conditions f)=2rx when <,r=1, 2, 3,. 1 1=1, 2,…

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Q: Show that f(u, v) = (u – 2) (v + 3) has neither local minima nor a local maxima at (2,-3). %3D

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Q: what local linearizatoin of the function f(x,y) sin(xy) + cos(x/y) at (pie/4, 1)  (show all steps…

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Q: bot A de groutne conitant. est Cheafuntis uh that f cark3 and f'4)=-s. g(xl=In (f(x)) + A conx)

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Q: Compute F. dr where F (x, y) : (7xy + x cos x) i+ (-yev + 6x²) j and C is a piecewise closed curve…

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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(x, y) = (- (3x + y))'. Then %3D drðy af

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Q: 5.) Use linear approximation to estimate f(0.1, −0.9) x² sin x In(y² + 1) Y f(x, y) = of th + x+1…

A: NOTE: Refresh your page if you can't see any equations. . here we want to find the approximate…

Q: Show that F = (2x - 3)i - z j + (cos z)k is not conservative.

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Q: ) Compute f-(, ") and fy(§, T) of f(r, y) = e9 cos(ry)

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

Q: Let y 4x + 3x + 4. If Ax 0.3 at a = 3, use linear approximation to estimate A %3D Ay =

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Q: Evaluate F. dr where F(x, y, z) = (3.75yz cos(xyz), 3.75xz cos(ryz), 3.75xy cos(xyz)) and C is 1 1…

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Q: Suppose that z = f(x, y) is differentiable at the point (27, 81) withf,(27, 81) = 2 and f, (27, 81)…

A: Given that:                            z=f(x, y)fx(27, 81)=2fy(27, 81)=-6x=t3y=t4

Q: Differentiate y= Vsin x. Vain x A * (xcotx-Insinx) Van Nsin x 810 X B. (xcotx + Insinx) Kainx -…

A: Here we have to differentiate     y = (sinx)(1/x) .

Q: tor. f(x,y)= In (8+3x²+3y²); P(3,-1); (3.2) d the gradient of f(x,y) = In (8+3x²+3y²). xx,y)=

A: Topic:- application of derivatives

Q: Suppose f is differentiable at (3, 4), V f(3, 4) = (-V3, 1), /V3 and u = Compute D„f(3, 4). 2

A: ∇f3, 4=-3, 1, u=32, -12

Q: Show that if |sin z| is bounded along the straight line {(x, y) : ax - By = 0}, then a = 0, i.e.,…

A: We know, sin z = sin x cosh y + i cos x sinh y, Where, cosh and sinh are hyperbolic functions.

Q: /alue of divergence of function (x2y2)î - (z³y – 4x) ĵ + 4y2 xk at (1, 2, 1) is

A: Let f = (x2y2)i^ - (z3y-4x)j^ + 4y2xk^

Q: Let y = 3x + 2x + 4. If Aæ = 0.4 at x = 1, use linear approximation to estimate Ay %3D Ay 2 Submit…

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Q: If x'y²z3 + zx sin y = 5 defines z as a function of x and y, find . O A. 3x2y2z² +x sin y 2xy2z3 + z…

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Q: 2. Suppose that a function f: R3 → R³ defined by: f (х, у, 2) 3 (2х + Зу — z, sin(тхy?z-1),…

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Q: that F(x, y, z)= Divergence Theoren

A: The given vector field is Fx,y,z=xz2+12.13, sin y, x·y5. The upper hemisphere can be taken as…