(b) Show that, the function u = In(x² + y²) + 2 tan" satisfies Laplace's equation.

Q: 2. Find 02z дхду for z = x2 tan-1 Y X

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Q: (4) Find the equation of the tangent line of y = /1+4 sin x at the point %3D (0,1).

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Q: ƏT ƏT and dr 3. Find of function T = x²y – xy³ + 2; x = r cos 0 and y=r sin 0

A: We have to find derivative

Q: Find an equation of the tangent line at the given point on the curve. (a) x = 2 – 3 cos 0, y = 3+ 2…

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Q: Find the derivative dydx of the polar function r=1+sinθ at θ=π4.

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Q: What is the point of intersection of the tangent lines to the curve r(t) = sin(πt)i + 8sin(πt)j +…

A: Let us consider the given curve r(t) the tangent line is obtained as Find the derivative of the…

Q: 3. Show that the complex function f(z)= cscz =1/ sinz satisfies the Cauchy-Riemann condition. Find…

A: The given function is fz=cscz=1sinz. A function fz=ux,y+ivx,y, satisfies Cauchy Riemann equations…

Q: d dx d du d dz d [12 cos ¹2x] dy d dq [4 sec ¹3u] [10 tan-¹2z] [6 csc-¹4y] [9 cot-¹39]

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Q: Find the tangent plane to the equation 2 = 4y cos(3x-5y) at the point (5, 3, 12). 2 =

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Q: Find the parametric equations of the tangent line LT to the curve C:r (t) = 2 cos(t)i + sin(t)j +…

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Q: nd the points at which f(x) = cos² x has a horizontal tangent line. %3D

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Q: (6) Find дл ду : ye+sin(2)=423 (8 points)

A: Explanation is given below:Explanation:

Q: Find the equation of the line tangent to the cardioid r = 1+ sin(0) at Ô0 =

A: Given that,, Here, differentiate w.r.t. ,we get, We know that, To find: equation of the line…

Q: Find stationary points of y = cos (4nt)

A: given function is

Q: Find the domain of the vector function 7(t) = Domain: {t| <t< }

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Q: 7. Use implicit differentiation to find two equations between x and y such that the tangent line to…

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Q: Find the tangent plane to the equation z = 5y cos(4x - 6y) at the point (6, 4, 20). z =

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Q: satisfied

A: We will solve the problem.

Q: Find the unit tangent vector to the curve defined by F(t) = (5 cos(t), 5 sin(t), 2 sin²(t)) at t = 5…

A: