Prove that.  lim(x→∞) (ln(x)/x^p) = 0   For any number p > 0. This shows that the logarithmic function approaches ∞ more slowly than any power of x

Q: Determine the first derivative of the ff. function y = tan(ln(cot(x² + 3 ln 3x)))

A: Chain rule of differentiation: f(g(x))'=f'(g(x))g'(x) Formulae used: ddxtanx=sec2xddxlnx =1xddxcot…

Q: By using the definition of hyperbolic functions as exponential functions, show that sinh 3x = 3 sinh…

A: handwriting solution

Q: How is the natural logarithm function defined as an integral? What are its domain, range, and…

A: Explanation:

Q: Use linear approximation to estimate ln(.99)

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Q: Give 1 problem for differentiation rule of exponential

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Q: ln(x) is just defined for x greater than 1. Select one: True False

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Q: find the length of the curve y = arcsin e^(-x) from x = 0 to x = ln 5/4

A: Given curve, f(x)=arcsin e-x from x=0 to x=ln54

Q: Find the limit of: limit of log5(8x-x4) as x approaches 2- (Use left hand limit)

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Q: Compute sinh(ln 5) and tanh(3 ln 5) without using a calculator.

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Q: What is the derivative of the natural logarithm function ln x? How does the domain of the derivative…

A: Given: fx=ln x To find: (a) The derivative of f(x)  (b) Compare the domain of the derivative with…

Q: Discuss the relationship between the derivative dy/dx and (Lim) with drawing if it is required ?

A: See the attachment

Q: By using Mean Value Theorem, show that ln(1+x)  0

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Q: By using the exponential definitions of sinh and cosh. show that cosh (t) - sinh (t)

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Q: 5) Find the derivative using logarithmic differentiation: y = Xcanx

A: As per our guidelines we are suppose to solve only first question. Kindly repost the other question…

Q: Find the derivatives of following fucntion (logx)tanx

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Q: 1 ソ= 1- x

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Q: Show that e grows faster as x –→∞ than x" for any positive inte- ger n, even x'-000,000. (Hint: What…

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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 derivative of the function by the limit process. f(x) = 6/x^2

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Q: Differentiate the following function.   y = ln(3x+1) - ln 3

A: We need to differentiate the following function.   y = ln(3x+1) - ln 3

Q: State the derivative rule for the logarithmic function ƒ(x) = logb x.How does it differ from the…

A: Given: fx=logbx for finding derivative of given function, first we convert it's base in e then…

Q: What are the domain and range of ln x?

A: Given: y=lnx we know that in lnx, x>0 so, domain of given function is x>0

Q: In(tanhx)

A: As we know that; The chain rule: fgx'=f'gx·g'x And ddxtanhx=sech2x Given: fx=lntanhx

Q: Find the absolute maximum value of ƒ(x) = x2 ln (1/x) and say where it occurs.

A: We need to find the absolute maximum value of the function

Q: Use the sign of the derivative to establish the inequaltiy ln(1 + x) > x − x2 2 , ∀x >0

A: SOLUTION-

Q: compute the derivative g(x) = tan−1(ln x)

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Q: -e lim te Int - 1

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

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Q: Find the indefinite integral.

A: Use the substitution method and consider x2 as u. Differentiate both sides of u=x2 and substitute x2…

Q: Compute the derivative of Sin x in two ways: (i) Using the chain rule. (ii) Taking the derivative of…

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Q: Find dx 1- y = log (Sin x )- Cot - Vx 3 2. y = log( vx) = tan(x - y) 3- x Sec -x = y? log (sec x) 4-…

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

Q: Find the derivative using logarithmic.   sq root of ((x2-1)/(x2+1))          for x>1

A: Here given for x>1

Q: give a problem and solution for differentiation rule of exponential and logarithmic functions in…

A: See the details solution in below

Q: Find the arc length: f(x) = (x³ + 5)ž – log x? from x = -1 to x = 3

A: Given:          f(x)=43x3+552-logx2       from x=-1 to x=3   Arc length =∫ab1+f'(x)2dx    here a=-1…

Q: If x >0, Prove that X - ln (1+x ) > x^2/2+2x

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Q: * limt (x-sinx)\ x+1 10 log x^2+3 O 0.5

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Q: se limit methods to determine which of the two given functions grows faster, or state that they have…

A: Let f(x) = ex2 , g(x) = e10xIf limx→∞f(x)g(x)=∞ , then f(x) grows faster than g(x)If…

Q: If a function is increasing everywhere (and it has a derivative), then it's derivative function is…

A: explanation of above question is given below,

Q: 1- Y= sin( coshx log2 2 ) 2-Y= sinx.coshx. log2 2 3 - y = In Vx. sin x

A: Apologize  that we are permit to do first question of the multiple questions but still I am doing it…

Q: if X #0 メ=ロ Frnd continuous

A: given the function fx=xx if x≠00   if x=0 we know that x=x if x>0x=-x if x<0 therefore,…

Q: how do you express lim,,→ Li=1 Σ! as п 1+ definite integral

A: We know that

Q: Find the derivative of f(x) Give answer in exponential notation with positive exponents

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Q: The L'Hopital rule and The derivative of the exponential function. lim (ex + x³)*:

A: Use l hospital rule

Q: Find the derivative using logarithmic differentiation. x(x + 4) (5x + 1)(3x +7) (Express numbers in…

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Q: Find the indefinite integral ∫ x2 sech2 x3 dx

A: We have to find indefinite intrgral  ∫x2sech2x3dx We have ∫x2sech2x3dx  Take x3=t⇒3x2dx=dt⇒x2dx=13dt…

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

A: To differentiate the given function.

Q: Find the derivative of the function. y=arctan(sqrt 1-x/1+x) Also, Would I be able to use…

A: Given :  y = arctan1-x1+x

Q: Find the derivatives of x log x.log(logx)

A: We have to find the derivative of-  x logx.log(logx)