Show that r = (sin t)i + (cos t)j + V3k has a constant length. And r(t) is orthogonal to it's derivative. %3D

Q: 5. Determine the equation of the tangent line to (a) r= 2 sin e at = (b) r =1+ cos 0 at 8 =

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Q: What is the equation of the line tangent to the curve e" + cos(x) = 1 at the point C.0. y =1- ... A…

A: Given; ey+cosx=1 at point (π/2,0).

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: Use the Chain Rule to find the derivative of w = xy with respect to t along the path x = cos t, y =…

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Q: Show that the function f(z) = e* – y?(cos(2xy) + isin(2xy)) is entire and find its derivative

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Q: Find the directional derivative of the function in the direction of u = cos(8)i + sin(8)j. fx, y) =…

A: Please note that according to guidelines only first Q is solved.

Q: Let f(x)= 9 sin x/ 2 sin x + 4 cos x Then f'(x)= The equation of the tangent line to y=f(x) at a =…

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Q: Find dy/dx by implicit differentiation. (a) Vry = x- 2y +1 (b) y³e2 + 53y = sec(3x)) (:) (b) x = In…

A: To find the implicit differentiation.  We use the , basic product and chain rule of differentiation.…

Q: Find the derivative d cos t dt dz

A: Given : ddx∫0xcostdt

Q: Find the derivative of y with respect to t at t=2 if a) у %3D х + sinх - cos(1 + x+x?) аnd x = (t/t…

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Q: Compute the derivative for r(t) = (cos(8t))i + (sin°(t))j + (tan(6t))k.

A: Derivative of r(t) = cos (8t) i + sin9(t) j + tan (6t) k

Q: Find dy/dx for e 2x + y 2 = sec(2x) in terms of x and y using implicit differentiation

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Q: 2x g(x) = |VI+t? dt 1+t2 dt %D

A: To find the derivative

Q: 2. Calculate the derivative dy for the parametric equation x = t² + ³ + ln(4t+1), y=e5t² giving your…

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Q: Find the derivative of y with respect to the independent variable. y = 10cos T) -.... A. - 10coS TO…

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Q: 2. Find equation of the tangent line at the point P given by t = where x(t) = cos (t) %3D y(t) =…

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Q: Find an equation of the tangent line of y = e" cos T, at z= 0.

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Q: Find an equation of the tangent to the curve at the point corresponding to the given value of the…

A: As per our guideline we are allowed to answer only one question. Kindly repost the other question as…

Q: Find the equation of the tangent to the curve at t = 0

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Q: Find the derivative dy/dx of the cardioid given by the equation r=f(θ)=a(1+cosθ).

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Q: Find the derivative of this complex function: where z=x+jy (2+jj?

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Q: find the derivative of y with respect to x, t, or u,as appropriate.55. y = ln (cos^2 u) 56. y = ln…

A: Since you have posted multiple questions, we will solve only first question for you. To getremaining…

Q: Find the derivative f (t) = 1- cos t | a. 1 cos t - 1 sin t C. 1– cos t - t · sint (1 – cos t)? b.

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Q: Q1: a) Find the directional derivative of S = In(2x +3y) at point (-1,1) in the direction of 4i – 3j…

A: (a) Given S=ln(2x+3y) Differentiate S partially with respect to x and y Sx=22x+3y…

Q: Find the linear approximation to h(y)=sin(y+1) at y=0. Use the linear approximation to approximate…

A: Recall that the linear approximation of a function y=f(x) at x=a is given by y=f(a)+f'(a)(x-a) Here…

Q: Assume that y= sin dy 'x is a differentiable function of x. By differentiating the equation x= sin y…

A: Given,           Assume that y=sin-1x is a differentiable function of x. By differemtiating…

Q: Let g(t) = (3t)Sin t, Find the derivative of g. %3D sin t g'(t) = (3t) sin. (21)] + cos t· In(2t)…

A: The given function is g(t)=3tsin t

Q: Determine the equation of the curve whose tangent line at any point (x, y) is e^- sin x and passes…

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Q: Find the derivative with respect to x. (c) y = v7x – x4 (d) y = (1+ sin 7.x)-5 (a) y = (5x – 2)*…

A: This question is related to derivative, we will solve it using given information.

Q: By referring to implicit differentiation techniques, compute dy at point dx (3, t) for the following…

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Q: What is the derivative of the following? sin(t) d dt f 117 da cos(t) s(t) 4-sin² (t) 1 4 — sin² (t)…

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Q: Use the Chain Rule to find the derivative of w = xy with respect to t along the path x = cos t, y =…

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Q: Q6:- Show that R = (sin t)i + (cos t)j + v3k has a constant length. And R(t) is orthogonal to its…

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

Q: 13. Find the derivative of the function T(z) = dz +b cz +d numbers such that ad – bc + 0. When is…

A: Hello. Since your question has multiple parts, we will solve first question that means question (13)…

Q: Compute the derivative for r(t) = (cos(8t))i + (sin°(t))j + (tan(6t))k. %3D dr dt

A: Given

Q: Calculate the derivative of the function r(t) i+ cos' (61)j + lIn (2t³)k. t - 2

A: Given : r(t)=-5t-2 i +cos-1(6t)  j +ln2t5 k We have to calculate the derivative of the given…

Q: The derivative of y =e' ,In(sin 2x) is a) y = eln(sin 2x) sin 2.x b) y = e a(sin 2x), cos 2.r c) y'…

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Q: Find the equation of the tangent plane to z = ln(2x + y) at −1,3

A: Find the equation of the tangent plane to z = ln(2x + y) at −1,3

Q: Find the slope of the tangent line to the path of a carnival ride x 7 cos() + sin(7t), y = 7 sin() +…

A: Given  x=7cos(t)+sin(7t) y=7sin(t)+cos(7t) We need to find the slope of tangent at t=0

Q: Find dy dx for sin(xy) + y³ = x² + 1 and use it to find the equation of the tangent line at the…

A: use differentiation for finding the equation of tangent line.

Q: Find the 3rd derivative of the function given by the parametric equations x=1+t2, y=t−t3 at t=1.

A: Differentiate x and y with respect to t.

Q: Q1: a) Find the directional derivative of S = In(2r +3y) at point (-1,1) in the direction of 4i -3i…

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Q: Find an equation of the tangent line to the curve at the point corresponding to the given value of…

A: Given that, x=tcost, y=tsint, t=π. Find the value of x and y by substituting the value of t in given…

Q: Find the directiona derivative of f (x, y) = In(e* – 4y at the point (0, 0) %3D - in the direction…

A: We have to solve

Q: We know that x =r cos 0 and y =r sin 0, then find the derivative af /ər if 1 f = x2 + y2

A: Given f = 1x2+y2  and x = r cosθy = r sinθ Thus f = 1r cosθ2+r sinθ2= 1r2 cos2θ+r2 sin2θ=…

Q: a) Differentiate implicitly to find for the equation x cos y = x2 y2 +1 dx %3D b) Find the equation…

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Q: 17. 2y° +4x2 - y =x° ² - y=x` +sin(3x)=12–j 4

A: We need to find implicit differentiation of : 17. 2y3+4x2-y=x6 18. 7y2+sin(3x)=12-y4

Q: Find the derivative of the function. y = 6 In sin 22x O A. 24 sin 2x OB. 12 In sin 2x C. 24 cot 2x O…

A: We can find the derivative on the given function by using the chain  rule of differentiation.

Q: 7) Let x = t – sin(t) , y = 1– cos(t). Determine the equation of the line tangent at t = n.

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Q: Find the derivative of f(x,y) = xe² +cos(xy) at the point (2,0) in the direction of V = 3i – 4 j .

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