D. 1. If (x + y)² = x² + y² 2. If y = sin³ () cos³ () dy - dx X prove that y' = sin² (x) cos(x) 8 prove t that

Q: 3. g(t) = arcsin (E - find g'(t) A. (4-t+) В. (4-t+)ž C. - 2t5 (4-t4)ž 2t D. (4-t+)5

A: 3-Given- gt=arcsint22-t24-t4 To Find- The value of g't.

Q: 4. Find (f-1)'(1) if f(x)=x' +8x+sin(3.x) а. Find (f-1)'(2) if f(x)= Vx' +x +xr+1 b.

A: To find the derivative of the inverse of a function

Q: 3 sin z If f(2) = then 2 + cos z f'(2) =| f'(5) -

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Q: Let f(æ) = sin(x) and g(æ) = x² . Find (go f)'(x) O 2sin(x)cos(x) O 2xcos(x?) -2æcos(x?) 2xcos(x)

A: Given that f(x)= sinx and g(x)=x2. And we find (g0f)'(x). We know that d(xn)/dx = nxn-1.     Here i…

Q: V. Show that: 1. [(cosx-sin x)'dx = x+ cos 2x +C 2.

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Q: (A) Evaluate Sin (3x) Sin (3x) dx

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Q: 68) Find a value of c that will make sin2 4x 0 x2 f(x) = C, x 0 continuous atx = 0.

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Q: One of the following statements is False. Is it a, b, c, or d?

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Q: 28. Show that tan z = sin(2x) + i sinh(2y) cos(2x) + cosh (2y)

A: We can show the given identity.

Q: Let f(x) = Then f'(x) = π The equation of the tangent line to y = f(x) at a = can be written in the…

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Q: f(æ) sin(x) T Given f 5 and f' 6. Also let g(x) f(x)cos(x) and h(x) Answe the following: T a) g' ()…

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Q: If z = x2 + y² and x t cos 2r and y = r sint, then %3D a2z is at2 (A) 2 cos²(2r) + 2r? cos?(t) – 2ry…

A: Given  z=x2+y2      and x= t cos 2r ,y= r sin t First substitute x,y values in z  Then derivative…

Q: 4. J (cos x – sin x)²dx

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Q: tan(): '(), Suppose z = x = u cos(5v) , and y = u sin(5v) . Compute dz/du and az/dv in two different…

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Q: 2. [ * xkbr cos*x- sin 3(sin y- cos y)° dv 10. cos y

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Q: 8. Differentiate with respect to x: (a) y = sin(x³), (b) y = sin³ x and (c) y = (7- cos(5x)) 2023

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Q: Find g'(0) if g(0) = cos 0 sin 0. [A] cos²0 [B] cos²0 + 1 [C] cos²0 sin²0 - - [D] sin²0 [E] sin²0 +…

A: As we know that; The Product rule: f·g'=f'·g+f·g' Use the common derivatives.…

Q: a) Suppose that f(4)=3 and f'(4)=-5. Find g'(4) where g(x)=Vx f(x) b) Let y=cosx-sin.x , show that…

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Q: Read each of the two statements carefully. Choose A(if both statements are true), B(if the 1st…

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Q: Let f(x) = cos(x). (a) On [0,2n) solve f(x) = 0. 4.

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Q: d) Bessel function, Show that So Jo(x) dx = 1.

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Q: 5. Let F(x) : = a sin(3.x) + bx cos(3x). Find constants a, b such that F'(x) = 2x sin(3x) and verify…

A: The detailed solution is given below Thank you

Q: Let y = sin (x + y), find y'.

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Q: 3) Show how the substitution u = 2 x shows that - - t/2 -Tt/2 S sin"x dx = S? cos"x dx

A: Let u=π2-x Differentiating u w.r.t. x- dudx=-1du=-dx Considering left hand side of the given…

Q: 2 sin(x) a + va? - 1 + cos(x) arctan Va? 1 Va? – 1 - - 1

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Q: Let y= Sin x + 2 (05 X ) Ivaluate y, y'sy" and y'" at 4

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Q: U tt = quna %3D u(x, o) = Cos(x) Ut (X, o) = los (2x) %3D

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Q: 4. Let f(x)= 5Vx and g(x)= 4 + sin x. (a) Find (f ° (x) and (f ° g'(x). (b) Find (g ºf)(x) and…

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Q: Q1) find dy/dx 1) F(x) = sin’2x tan x² x – 3 ) 2) F(x) = sin( x +4

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Q: dx √(1-x²) 7. The value of f A. CONNE Oc O J. N is B. sin '(x) + c 4 C.rosx + le D. cosx A 8. Jcos'…

A: I am attaching image so that you understand each and every step.