Given f(x, y) = sin(x + y) where x = s°t, y = 3s – 2t. Find f.(x(s, t), y(s, t)) = ft(x(s, t), y(s, t)) =| Note: This question is looking for the answer to be only in terms of s and t.

Q: Consider f(x,y) = cos(x² e³). Then which one of the following is correct O!. fx(x, y) = – 3 y² sin(3…

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Q: Q. If z = f(x, y), where x = u cos 0 – v sin 0 and y = u sin 0 + v cos 0, with 0 a constant, show…

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Q: Find fr(x, y) and fy(x, y) given f(x, y) = tan x· cos os(x² – y?).

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Q: Q:Ifx = sin 6, y = 0,y= cos p6, prove that (1-x )y, xy,+ p°y= 0, dy and y1 d dy where y, %3! %3! de

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Q: M(x,y) dx + (x,y) dy=0 to be exact

A: Solution :

Q: If f(x,y,z)=sin(xy)+x2lny, then fxy at (0,pi/2) is equal to 1 FO

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Q: If z = f(x, y), where x = r cos 0 and y = r sin 0, show that a²z 1 a°z 1 əz r дr дх? dy? дr?

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Q: The linear approximation at r 0 to sin(2r) is A+ Br where A is: and where B is:

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Q: Find (∂z)/(∂s) and (∂z)/(∂t), given: (d) z = sin (4x + 5y); x = s + t, y = s - t

A: Given that, z=sin4x+5y ; x=s+t ; y=s-t Then we get, z=sin4s+t+5s-t⇒z=sin4s+4t+5s-5t⇒z=sin9s-t

Q: Find a. Əw dx a. aw əx and b. y (2π,1,-π) Əw at the point (x,y,z) = (2π, 1₁ - π) if w = x² + y² + z²…

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Q: Find (∂z)/(∂s) and (∂z)/(∂t), given: (b) z = x2 + 3xy + y2 ; x = sin s + cos t, y = sin s - cos t

A: Consider z=fx,y & x=fs,t, y=fs,t Then, the derivative of z with respect to s is given by:…

Q: B. and , if z = (x² + y²)³/2, and r = e-" sin v, y = e¬u cos v. В. %3D

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Q: (r, y, z) = sin 2x i + xz cos y j+ e*y iv F,

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Q: Given f(x, y) = sin(x + y) where z s3t0, y = 3s – 6t. Find T = f.(z(s, t), y(s, t)) = f:(x(s, t),…

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Q: - If x=sect and y = tan t, evaluate at t of the following: 4. dy (a) dx d y (b) dx?

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Q: 1. If H(y) = cos y – sin y, Find H (0), H (),H (1T), and also show that %3D + x) = H (T – x) = -H…

A: Given that H(y)=cosy-siny find H(0), Hπ2, Hπ Show that  Hπ2+x=Hπ-x=-H-x

Q: = f (x, y) is differentiable and x=u cos v and y =u sin v. Show that z = Oz Oz 1 + ди + и

A: According to question given that z=f(x,y) is differentiable x=u cos(v), y=u sin(v)

Q: given sin -In\x+ y + 19| Ax, y) = V18 – (x + 11)² – (y + 10)² sketch domain p(-11+ v7, - 10) is…

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Q: Find e + Uy, Given that u(x,y) tan(x+11Xy+11y") .t/ - *א- w

A: Given ux,y=tanx+11x2y2+11y411-x2-y2

Q: Find dz/dr if z as a two variables function in r and y is defined implicitly by y3 + x22 + r sin r…

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Q: Given f(x, y) = sin(x + y) where x = s°t2, y = 6s – 2t. Find f.(x(s, t), y(s, t)) = fe(x(s, t), y(s,…

A: From starting itself if we replaced x and y interms of s and t in f we can easily work in that.

Q: Let y, = sin(t) be a solution to t y" -y' +4 t y=0. If y(t) is the solution to the I.V.P ty" -y' +4…

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Q: <plain why y = t² – sin? t cannot be a solution to y" + p(t)y' + q(t)y = 0 nere and q are both…

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Q: If z = sin(rry). and * = t+ s, y = t2² + 8 8z when t=0, s=1. at Find

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Q: Let y, = sin(t) be a solution to t y" -y' +4 y=0. If y(t) is the solution to the I.V.P ty"-y' +4 t…

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Q: find dw/ar if w = (x + y + z)?, y = cos(r + s), at the point (r, s)3(1,-1) x = r - s, z = sin(r + s)

A: Given w=(x+y+z)2 x=r-s y=cos(r+s) z=sin(r+s) point (r, s)=(1, -1)   when we differentiate partially…

Q: S(z, y, 2) = 122 y sin(z) – Vy- z cos(r+y) 8PR Find Vf.

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Q: Given f(x, y, z) = xe"y + sin(xyz), find fyzz

A: Derivatives of: Sin x is Cos x Cos x is -Sin x u

Q: Given f(x, y) = sin(x + y) where x = s°t4, y = 6s – 4t. Find f.(x(s, t), y(s, t)) = f:(#(s, t), y(s,…

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Q: Given f(x,y,z) = cos(4xy) + sin(4z). Find fx(8,9,0). Leave your answer correct to two decimal…

A: Solution

Q: Given f(x,y) = ex cos y, Evaluate fx(2,0) A cos 2 B) sin 2 C) 0 D) e²

A: fx is the partial derivative w.r.t. x

Q: If y = 1+ sin t ,x = cost at t = Findy by using Chain Rule, then choose the correct answer: dt

A: Given that: y=1+sintx=cost

Q: If f(x, y) = x cos y + yex, find d2f/dydx.     −sin y + ex     −sin x + ey     sin y + ex…

A: Note the differentiation results : d(ex)dx=exd(xn)dx=nxn-1d(cosy)dy=-siny

Q: Q:Ifx= sin 6, y = 0,y = cos pe, prove that (1-x )y,-xy,+p°y = 0, %3D dy where y, dy and y1 d

A: Here given that x=sinθ, y=cospθ Taking derivative of both x &amp; y with respect to θ, we get…

Q: Given that f(x) = sin a cos' r +2 (sin r + a)' +, the value of the constant and the least value of f…

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Q: Given f(x, y) = 4e2" sin(2y) %3D then V f(0, 7) =

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Q: Given f(x.y) = e X cos y, Evaluate fx(2,0) e2 cos 2 O sin 2

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Q: Given f(x,y,z) = cos(5xy) + sin(4z). Find fx(6,6,0). Leave your answer correct to two decimal…

A: Given f(x,y,z) = cos(5xy) + sin(42). Find fx(6,6,0)

Q: Find fr(x, y) and f,(x, y) given f(r, y) = tan r - cos(r² – y³).

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Q: Given f(z, y) = 7x° cos (y*), find fzy(z, y) = | fyy(T, y) = [

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Q: • T/4 cos(0) f |/| f(x, y, z) dV = 9p sin(ø) cos(4) dp d0 dø, find f(x, y, z). E f(x, y, z) =|| ?

A: We have to given ∫∫∫Efx,y,zdV=∫0π2∫0π4∫0cosθ9ρ2sinϕcosϕdρ dθ dϕ Now the function which is to be…

Q: Given: f(x, y, z) = e¯ª sin(yz) Find: ² f(x, y, z) əyəx O z cos(yz) et O cos(yz) e O-ze-* cos(yz) O…

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Q: Find ∂w∂s and ∂w∂t when s = 1 and t = 2π for w = xy + yz + xz                    where x = s cos…

A: Given: w=xy+yz+xz where x=scos(t), y=ssin(t), z=t at s=1 and t=2π

Q: 4. Find дw b. dz a. дх at the point (x, y, z) = (0, 1, T) if w = x² + y? + z? and y sin z + z sin x…

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Q: So Evaluate F. dr where C F = (– 2z, – y, – 4x), and C is given by 7(t) = (sin(t), cos(t), t), 0 < t…

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Q: Given f(x,y) = ex cos y, Evaluate fx(0,2) O e² 00 O cos 2 O-sin 2 M

A: We have to find the derivative

Q: The linear approximation at I = 0 to sin(3r) is A + Br where A is: and where B is:

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Q: Given f(x, y) sin(x + y) where x = s*t, y = 4s – 2t. Find f.(x(s, t), y(s, t)) = fi(r(s, t), y(s,…

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Q: Given f(x,y) = ex cos y, Evaluate fy(0,2) A cos 2 B) e² CO (D) -sin 2

A: This question can be solved using the concept of partial derivatives.

Q: for w = x sin y +y sin x + xy. W yx Q2) A) Verify that wry

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