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Math problems

(See attached file for full problem description with proper equations)

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The problems need to be solved in full and to show all work

1a) Find the following derivative implicitly with respect to x
If,

Y=(1+xy)^(1/xy). Find dy/dx! without simplifying the derivative.

Compute dy/dx at (1, 1).

b) find the formula for nth derivative of y=f(x)= √x

2a) let f(x)=(1+cos x) ^ sec(x)

find lim f(x)
x Π /2

b) y=(xy)^1/xy, find dy/dx and simplify y'( y prime)
evaluate y' at (e, 1/e)

3a) find the equation of the linear approximation of f(x)= sin(x). for x, near zero

b) y=f(x)= (ln x)^e^(1/cos(x)), f'(x)=dy/dx

4a) use differential to approximate cos(11/36 Π), hint f(x)=cos(x), choose x o= Π/3 (xo x of zero)

b) evaluate √2, using Newton method for n=4 steps

5a) let y=tan(x) and y=2x graph both equation and find the value of x where the two graph intersect, using Newtons method n= 4 steps use xo=1.25

b) show that newtons method fails for y=x^1/3

xo=0.1

6a) the formula for amount A (maturity or future value in a savings account compounded n times a year for t years at interest rate of r and initial deposit p is given by the following formula below/

A=p{1+ r/n)^nt, find the formula for A if the number of compounding period per year becomes larger or n∞

b)find the derivative of the new formula in 6a) with respect to t

7a) lim (1 - 2/x)^x
x∞

b) lim (sin)^x
xo+

8a)if f(x)=x^3 +x-1 use, rolles theorem and intermediate value theorem to show that f has exactly one real root

b)
find the number c for f(X)=x-x on [0 2] where the instantaneous rate of c is equal to the average rate of change of f(X) on [0 2]

9a) Graph y=f(x)= x^x and find two critical numbers for f(x)
and determine if f is differentiable and continuous on both numbers you have found

b) determine the region in which f is concave up or down

10a) Let f(x)= 3x^4-16x^3+18x^2. find all critical numbers, the interval of increasing, decreasing, concavity, inflection point for problems ( 10a and b)

b) g(x)=x^⅔ (6 - x)^⅓

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This solution is comprised of a detailed explanation to find the following derivative implicitly with respect to x.

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