Homework: 1. If y=tan x, prove that y`= 2(y +1)(3y² +1).

Q: aw Example 3: Find at „w=2ye*-In z , x=In(t² + 1) ,y=tan-1t &z=et

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Q: Q6:-Prove that: y" =1 y =csc(tan"") if

A: Given y=x4·csctan-114x2-1 Let tan-114x2-1=α ⇒tan(α)=14x2-1 ⇒cscα=2x ⇒y=x4cscα⇒y=x4(2x)⇒y=x22

Q: 4. How many fixed points does the function T(x) = tan(x) have? %3D

A: for fixed point f(x) = x x = tan(x)

Q: (b) Let f(x) = sin (cos-1 (sin(cos-vx))).B f'(1) = using chain rule, show that %3D

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Q: dx Evaluate: J sin x A. 3tan'x ++C C. tan'x +C tanx tan x + C tan 2x D. в. 3tan'x tanx 2tan2x

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Q: = Assuming that y = - sec(xy). It will follow that y' Select one: True False y sec(xy) tan(xy) 1-x…

A: Note : As per our guidelines we are supposed to answer only one question if there are multiple…

Q: d Q 3. Prove that tanx dx sec?x. =

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Q: Let y = tan10 (x5).Find ƒ, g, and h such that y = ƒ(u),where u = g(v) and v = h(x).

A: The given function is

Q: If f(x) = cos a- 5 tanx, then %3D f'(x) = %3D f'(5) =

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Q: if y=tan x,prove that y'''=2(1+y²)(3+y²)

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Q: If y = sec (x* ), then is A. Ssee(x*) tan"(x*) B. 10r sec (x* ) tan" (x²) c*(x*) D. 10k tan(x*)…

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Q: du du If u = sin -1 x+y'0, Prove that x xty +y- +y==tan u. dx dy

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Q: In Exercises 1–6, find the gradient of the function at the given point. Then sketch the gradient…

A: 1) If F=F1i+F2j is any vector then gradient of F is, grad(F)=i∂∂x+j∂∂yF

Q: 2. S tan (x)sec*(x)dx = tan10x a. tanx + 8 + C 10 tan10x b. tan®x + C 8 10 sec10x с. sec®x + C 10 8…

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Q: 2.xy x² – y? Show that - = 0 if z= Tan

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Q: 1 Suppose y = tan ¹æ. Then x = y tan Otan y Oarctan y

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Q: bx + c Given tan 3 - In(x2 - 8x + 25) + 1 -dx %3D x2 - 8x + 25

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Q: 1 tan 2 1 1 + tan-x +C. 2 2

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Q: du nd az and " if z = 5 tan-'(x)and x = e" + In(v) at tl-

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Q: (x cos + y sin?)y – (vsin? -x cos?)* х — х COS = 0 Ans: xy cos = c (Lucknow, 1997; Kanpur, 1996)

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Q: Let a > 1 be a constant and y = (9e- sin z 3) In(x+a). Find y'(0).

A: Product rule of differentiation:…

Q: 4. How many fixed points does the function T(x) tan(x) have? %3D

A: As per our guidelines, we suppose to answer only one question. Here question 4) is answared.

Q: Hw) if J-0- Gso and X=O+ Cas e Show that Sece+tan 0

A: The given differentiation can be solved by using chain rule. Initially, the y expression can be…

Q: (B) The evaluation of f- -dx is (x*+3x2 +1)[tan-( In|tan"+5| + e -In tan-( +c (i) (ii) (iii) None of…

A: No antiderivative could be found within the given time limit, or all supported integration methods…

Q: Find the absolute extrema on the interval [0,2π] for f (x)=x/2+cos x.

A: Absolute extrema

Q: (i) Let f(x) E[an cos(n7x) + b, sin(nax)] , %3D where, | +x, -n <x< 0 | 0, f(x) = 0 < r<T. Then, b,…

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Q: If y = tan x, prove that y``= 2(1+y² )(3+ y²).

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Q: implicitly by the equation y tan-1(xz) − x cos(yz) = 3. Find zy.

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Q: J교+2x +5 글m() + 4| + tan-! C n() ++| - } A x+1 '(***) + c tan-! + C B !In(x + 1) + 시 + tan. x+1 D…

A: Divide and Multiply by 2 . And solve it.

Q: Let y(x) be the solution for the DE. y' + y tan(x) + sin(x) = 0 If y(0) = In 8, then at x = 4 the…

A: A first-order ordinary differential equation of the form y'(x)+p(x)y=q(x) can be solved using the…

Q: 12. Given that V3 cos x = cos(-x), tan x= A 3. cos(f B V3 C 1. V3

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Q: 7. Let f(x,y)= a) T 2 xsin y-y, find sin x+y Simplify and give exact value

A: NOTE: Refresh your page if you can't see any equations. . take the partial derivative with respect…

Q: dx - = In[F(x)]+C. What is F(3) if C= 0? For %3D (x+x³) (x Tan-!x+1) -1 A +

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Q: dx = In[F(x)]+C. what is F(/3) if C = 0? For (x+x³) (x Tan-'x + 1)

A: We will integrate the function first.

Q: 26. y'+y tan x = cos x; y(x) = (x + C) cos x, y(n) = 0 %3D 25. y' = 3x²(y² + 1); y(x) = tan(x³ + C),…

A: 25)y'=3x2y2+1---Iyx=x3+Cy0=0solution:yx=x3+C⇒y'x=sec2x3+Cddxx3+C=sec2x3+C3x2y'x=3x2sec2x3+C---Aputti…

Q: If y= tan x , prove that y`= 2(1+ y² )(3+ y²).

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Q: dx -In[F(x)]+ C. What is F( 3) if c = 0? For (x+x³)(x Tan-!x + 1)

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Q: 1) If dy = tan x, then y =? dx А. !tan?x + C B. sec?x + C C. In [sec x]+ C D. Inſcos x] + C 2) dx =?…

A: We have to choose the correct option.

Q: Solve the equation for x, where x is restricted to the given interval. y = 3 tan (2x - 1), for x in…

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Q: 5) Let y= cos' (S(x))+e". Find y' at x 1, given that f(1)= and f'(1) = 30

A: Given query is to find the derivative at indicated values.

Q: 47. Let y = sec-x. Show that tan y = vx? – 1 if x > 1 and that tan y = -Vx? – 1 if x 0 on (0, 3)…

A: Consider the function of x and solve further-

Q: 213 If cot 0+ tan = and sece - sece = y, Prove that (x'y)° (xy²)*° = %3D

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Q: dx = In[F(x)]+C.What is F(/3) if C = 0? For (x+x³) (x Tan-lx+1)

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Q: If y= tan x, prove that y 2(y² +1)(3y² +1).

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

Q: For sin y=- find the value of so that (y+y")= - 1 at 1 at x=0 Cos X-

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Q: Q6:-Prove that: y" = 1 y =csc(tan"" ) if

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Q: 6. Using a suitable substitution in each case, integrate the following w 1 (a) x2 + 10x + 29 sec?,…

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Q: : s ech[In cosx]tanx dx Evaluate: A None - 2 tan-1(cosx) + C 2 tan -1(e cosx) + C CoSx (D - 2 tan…

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Q: dy - y tan x = 10 tan x dx ution of y(x) using two different techniques. Hint: tan x dx = - In(cos…

A: We have to solve given problem: First method:

Fundamentals of Trigonometric Identities

Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.