1. f(t) = cos(6t) 2. f(t) = (e^(8t)) cosh(3t) 3. f(t) = (t^2) e^(-4t)

Q: Find the first derivative of 2cos(2 + x³). a. -6x² sin(2 + x³) b. -x² (sin 2 + x³) c. -6x² (cos…

A: daf(x)dx=adf(x)dx ; where , 'a' is constant. dcos (ax)dx =-a.sin(ax) dxndx=n.xn-1 .

Q: Find a curve through the point ( 1 , 5/6 ) whose length on the interval [ 1 , 2 ] is given by ∫ 1^2…

A: The length of the curve y=f(x) from x=a to x=b is L=∫ab1+dydx2dx Now , Length of a curve on the…

Q: Express in exact form: In(x² + 1)dx

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Q: 1) [(sinh 2x + cosh2x)° dx

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Q: By using the definition of hyperbolic functions, show that sinh x = 1 (cosh 2x – 1). -

A: Given,         sinh2x=12cosh2x-1

Q: Use the addition formulas to prove sinh(2x) = 2 cosh x sinh xcosh(2x) = cosh2 x + sinh2 x

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Q: One solution of a 2nd order homogenous LDE is e2*sinx. What is its general solution?

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Q: Find the derivative of: 3 sin (– 3x). Hint: sin (x) = [sin(x)]"...so use the chain rule (twice!).

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Q: Show that coshxt sinhx= ex and coshx- SinhX = e-x, Hen ce deduce cosh?x- Sinh?X=|

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Q: 2 cosh(2x) – sinh(2x) = 2

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Q: Find the first four derivative  if  y= cosx

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Q: One side of a right triangle is 150 m long and its adjacent angle is π/3 radians, with π/60 radians…

A: Possible error

Q: (b) Prove that 1+ cosh x + cosh 2x coth x sinh x + sinh 2x 1+ cosh 2x + cosh 4x dx Hence evaluate…

A: 4b. Given,

Q: sinh(9x) If g(x) : 8 + cosh(9x) find g' (x) 8+ cosh(0)

A: Following formulae are used to solve first question

Q: FInd the derivatives of the function f(x) from the first pribnciple  cos2x

A: Derivative from first principal is ratio of change in y to the very small chance in x.

Q: Show that Costy+ Co5' (-x ) = TT.

A: To show that the given identity is true,

Q: 4. Describes the various ways in which integration and differentiation may be viewed as inverse…

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Q: 6. y%= arcsin a Va2-x2

A: Given: 6. y=xa2-x2-arcsinxa

Q: Flnd * |(sin 5x cos 3x + cos 5x sin 3x) dx cos 8x cos 3x + 3 cos5x – + c 8 + c Option 3 Option 2 cos…

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Q: solve for the intergral of (2sinx)/(1+cos^2x)dx

A: The integral to be evaluated is  ∫2 sinx1+cos2xdx. Here, we can use substitution method to evaluate…

Q: dervivative of (ln(cosh(x)))3

A: Given data:     The given function is: y=(ln(cosh(x)))3. STEP-1 Assume u=coshx and differentiate it…

Q: he measurement of the radius of the end of a log is found to be 20 inches, with a possible error…

A: Given that: Radius of log, r=20 inError in measuring radius, dr=18 in

Q: : Find value of x if :- 2 sinh (2x) - 4 cosh (2x) = - 4

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Q: Compute coshx and tanhx, assuming that sinhx = 0.8.

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Q: Find ∫cosh 2x sinh2 2x dx.

A: To find the value of the integral∫cosh2xsinh22xdx

Q: logtan nx(2+ 4 cos nx) 2

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Q: Please kindly use chain rule on the below to find the derivative and label all parts.  h(x)= cos…

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Q: Find first and 2nd derivative guncic 2 tan2x 3.

A: Derivatives can be defined as rate of change of function with respect to an independent variable. It…

Q: 2. Integrate using Trig: cos° x dx

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Q: (7) Find if \.Y =ln12- Cosx) , 2- Y= %3D

A: Calculate dydx. 1. y=ln 2-cos x. Derivate with respect to x by applying chain rule: dydx=dydududx.…

Q: 8. The value of f sinh* x cosh³ x dx

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Q: Find the derivative OF the exponential funchion y = In ( 5- e **) Please simplify it Further.

A: Consider the provided equation, y=ln5-e4x Find the derivative of the equation. Apply the chain rule…

Q: uxiliary equation of + by = 0 are re tion of the different

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Q: Use differentials to approximate √16.5.

A: Given:

Q: 1 3), sinh(x)-cosh(x) dx

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Q: Find* esin ¤ dx tan x csc x cos x e $in xt. - cos x e Sin x + c Option 4 Option 2 esinx + c ,sin.…

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Q: Use differentials to estimate sin (π/6 + 0.01).

A: Use differentials to approximate the value of sinπ6+0.01 The nearest number to sinπ6+0.01 whose sine…

Q: The first derivative of the function y = 3x cannot be solved using implicit differentiation. Select…

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Q: Find the possible values of cosh x that satisfies sinh²x – 3 cosh x + 3 = 0

A: We use the following identity to solve question 1). cosh2x - sinh2x = 1

Q: 2. One side of a right triangle is known to be 150 cm long and the adjacent angle is measured as…

A: The given problem is to find the maximum possible error in area of triangle when one side and angle…

Q: 4. Use implicit differentiation to find az in xexyz sin(2z) – xyz = 5. ду

A: I am going to solve the given problem by using some simple calculus to get the required result of…

Q: (b) Prove that : - 1) (coshx)2 - (sinhx)2 = 1 ,2) coshx - sinhx = e, 3) (cothx)2 -1= (cschx)2 %3D

A: Use hyperbolic  function

Q: second order derivative of (5 sin 3x+cos^2 (3x )) is *

A: The function: y=5 sin3x+cos23x

Q: if 2 tanh x.In x = sinh' (sin x)+logcos dx dy Q5/ Find

A: To differentiate the given function.

Q: Suppose we are asked to compute the 100th derivative of x 3 . How can we obtain the answer in a fast…

A: Solve the following

Q: 12. f(x) = ( cosx 1+sinx)

A: f(x)= (cosx/ 1+sinx)2

Q: In estimating | cos(4x)dx using Trapezoidal 1 and Simpson's rule with n = 6, we can estimate the…

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Q: Solve the first and second derivative of y = cosh2x+ sinhx

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Q: In( cosh(3x) - sinh(3x) )=

A: