Let f(x, y) = In(2x + 3xy) then 1 fr(1,1) : 2 fy(1,1) = = 5 3x fy(x, y) = 2x + 3xy

Q: If f(x, y, z) = ln(xª + yl0 + 2³) + V (x10 + y8 + z4), then find fr, fy and f; at (1, 1, 1).

A: The given function is fx, y, z=lnx4+y10+z8+x10+y8+z4 and the given point is (1, 1, 1). Evaluate fx…

Q: Let f(x, y) = ln(x + y − 1). Then, ƒ(e, 1) =

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Q: Compute DF(, ) of F(1, y) = (e" cos(ry), r² + y) %3D

A: Given function is F(x,y)=eycos(xy), x2+y Let  f1(x, y)=eycos(xy)f2(x, y)=x2+y

Q: f(x.y) =e^sin(y) + In(xy) ; әз of Әхд-у

A: Given: fx,y=exsiny+lnxy To find the partial derivative of ∂3f∂x∂2y Partial differentiation: Partial…

Q: Let f (x, y) = 2x3 + x2y - y2. Compute d2f/dx2, d2f/dy2, and d2f/dxdy at (x, y) = (1, 2).

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Q: Sx, ifx 1 f(x) = Show that f is R- integrable.

A: R- integrable function

Q: Find fr, fy, and fz. f (x, Y, z) = z In(x°y²cos(52)) fa = fy = fz = ||

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Q: Find d3f where f(x, y, z) = ln(x + y − z) + cos(x2y)

A: As per the question we are given a multivariate function as : f(x, y, z) = ln(x + y − z) + cos(x2y)…

Q: Find fr and fy- S(x, y) = /"9e" dt f.(r, y) : %3D f,(x, y) =

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Q: Find the partial derivative of f (x, y) = e2+3y cos (x³y²) with respect to x. O fa (x, y) = e2x+3y…

A: Given f(x,y)=e2x+3ycosx3y2 To differentiate f(x,y) partially with respect to x Let u=e2x+3y…

Q: Evaluate fSS 6e(²+y²+z²)/² dV over B where B= {(x, y, 2) : x² + y² + z² < 1}. Take T = 3.

A: Since we have    x2 + y2 + z2  expression in the integral and region B is the sphere of radius 1…

Q: 2, 0 1

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Q: Find fr, fy, and fz. f(r, y, 2) — вуzIn(4ry) fz fy f: =

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Q: Let f(x,y)=3x2-2xy+y2-8y  Find the absolute extrema over [0,1]x[0,2]

A: Consider the given function fx,y=3x2-2xy+y2-8y First, find the partial derivatives of the function…

Q: Let f(x, y) be differentiable at (ro, yo). Then we have f(xo + Ax, yo + Ay) - f(xo, y0) fa (ro, yo)…

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Q: \+etari), find f.. S, and f.- xyz 3. If f(x,y,z)= hn 2

A: We have to find fx , fy , fz.

Q: For f(x,y,z)=3xy+2y+8z - cos(x²+y²–z), the partial derivative fxy(1,2,5)=

A: Given function is fx,y,z=3xy+2y+8z−cosx2+y2−z. We have to find the partial derivative fxy1,2,5.…

Q: Let f(x, y) = ln(x + y − 1). Then, ƒ(1, 1) =

A: The given function is;  f(x,y)=ln(x+y-1).

Q: Compute the given partial derivative. f(x, y) = 8 sin(4x² – y), (0, 1) f,(0, 1) = %3D

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Q: Find fa, fy, and fz. f (x, y, z) = 3z In (x°y³ cos(4z)) fr = %3D fy = fz =

A: Given that , fx,y,z=3zlnx6y3cos4z find fx , fy and fz.

Q: What is the integrating factor of the linear DE of order one ydx + (3x - 2y +5) dy=0,

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Q: Find fr, fy and fz. f(x, y, z) = x – Vy² + z? |

A: Given f(x,y,z)=x-y2+z2. The objective is to find fx,fy and fz.

Q: Find fæ, fy, and fz. 4.5 f (x, Y, z) = 4z In(x*y° cos(5z))

A: To find the partial derivatives with respect to x ,y ,z.

Q: Find fr and fy- f(x, y) = 7e dt f-(x, y) = f,(x, y) =

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Q: Find f, (x,y) and fy (x.y), and explain why f(x,y) has no local extrema. f(x.y) = 8x + 3y + 4. fy…

A: It is partial derivative

Q: If f(x, y, z) = 2x5 + 3y5- 8x2y?z?, what is the partial derivative of f with %3D respect to z? fz =…

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Q: Find fr and fy. 2e" dt f (x, y) = 2, f.(r, y) = f„,(x, y) =

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Q: Let z = f(x, y), x = rcos 0 and y = r sin 0. Evaluate the value of zg at the point (x, y) = (/3,1)…

A: Chain rule: If is function of and and are functions of and then, Given:

Q: Find the linearization of the function f (x)= x³ at x = 1. qb |a| sin (a) L (x) = Using this…

A: Given, fx=x5 We know that, If we have a differentiable function f defined near "a" the linearization…

Q: Let f (x, y, z) = ln(x2 + z) + y2 exz. Find fzyx

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Q: Find fa, fus and fz. f (x, y, z) = (5z)) 3z In (x*y cos

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Q: Calculate fw(-1, 2, 0, –1) where f(x, y, z, w) = xy²z – xw² – e2a²-v²-²+2w²

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Q: Let f (x, y) = 2x³ + 4y³ + 19xy Find D = frx · fyy - fey at the point (2,2)

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Q: Let f(x, y) = 3e" sin(5y) af af dy

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Q: Find fr and fy. f (x, y) = 4e" dt f(x, y) = f,(x, y) =

A: A detailed solution is given below.

Q: Find fa, fy, and fz. f(z, y, 2) = z In (a*y cos(52)) z In (a*y* cos(52)) fr = fy = f2 =

A: First order Partial differentials

Q: Find fr, fy, and fz. f(x, Y, z) = 4yz In(6ry) fz = fy = f: =

A: Given: f(x, y, z)=4yz ln(6xy)

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

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Q: Find fa, fy, and fz. f(x, y, z) = 5z ln(x³y³ cos(z)) fx = fy fz f₂ = =

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Q: Find fr, fy, and fz. f (x, y, 2) = z In(x*y³ cos(52)) 4.3 fa = fy = fz =

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Q: Find fr and fy- f (x, y) = | 5e dt %3| fa(x, y) = fy(x, y) =

A: We apply fundamental theorem of calculas and solve it.

Q: If f(x, y, z) = In(x³ + yl² + 2®) + V (x12 + y6 + 23), then find fr, fy and f; at (1, 1, 1).

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Q: The partial derivative of f(x, y) = e"y – 2x – 4y + 3 with respect to x is O ye"V – 2 af af ery + y…

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Q: f (x, y, z) = x² + y² + z² and u = (1,0, 1) at P(1, 1, 1).

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Q: (ii)If y=(x+Jx* +1. prove that y'= Vx +1

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Q: Find the linearization of x = -1 f(x) = 3+ S sec(t – 1)dt at

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Q: siven f(x,y,z) = sin (3xy) + 3xyz3. Find fxzy(3,2,2).

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Q: 2xy F(x, Y) =Vx+ 2y (1, 1) f,(x, y) = 1,(1, 1) S(1, 1)

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Q: Find fr and fy- f(x, y) = fetlo dt f.(r, y) = fy(x, y)

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Q: F(x, y, z) = (xy — 2², 2x+cos(yz), 2y² - sinz)

A: Given the vector field F⇀x,y,z=xy-z2,2x+cosyz,2y2-sinz . The objective is to find the divergence and…