Let X ~ exp(rate = 3). (a) Find the mgf of X. (b) Use the mgf to find E[X], Var(X), E[X4] and E[X").

Q: Determine the first derivative of the ff. function dz y = e? sin x + In x (arcsin z) with dx 4

A: Determine the first derivative of the function                   y=ez sin x + ln x (arcsin z)  With…

Q: Find the differential of the function: y = x In x

A: Solution:Giveny=x ln xTo find the diffferentiationformula:(uv)'=uv'+vu'ddxxn=nxn-1ddxln x=1x

Q: What is the critical point of the function f (x) = cos x?

A: what is the critical point of f(x)=cos(x)? for critical point, f'(x)=0

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Q: Find the average rate of change of f(x) = cos x from 0 to π /2 .

A: Given: fx=cosx  0,π2

Q: Find the derivative. h(x) = In(cosh(7x)) h'(x) =

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Q: What is the average rate of change of y = cos (2x) on the interval [0,π/2]

A: Given f(x)= y=cos (2x) to find the average rate of change over the interval (0,pi/2) average rate of…

Q: Find dx 2 using implicit differentiation: In x + In y + e tan x = 0

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Q: Use the General Power Rule, Exponential Rule, or the Chain Rule to compute the derivative. y = cos…

A: Given that y=cos6θ+15 Find dydθ Needed formulae: xn'=nxn-1 cos x'=-sin x ax'=a

Q: x + x dx x2

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Q: find the second derivative of y=ln(e^x+e^-x)

A: Given query is to find the second derivative of the expression.

Q: Find the rate of change of y with respect to x at the indicated value of x. sin(x) :- 11 - cos(x)' y…

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Q: Find the second implicit derivative of the given using implicit differention y2-3x+2y=0

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Q: Compute the derivative of the function y=x2/ln x at x=e.

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Q: Find dy/dx by implicit differentiation   x2 + 5x + 9xy - y2=4

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Q: Find the first derivative of: y = sinh e2" O y e2 cosh e2# 2e2" cosh e2" O y' 2 cosh e2 O y = cosh…

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Q: Let f:[-n, 7] –→ R, where f(x) = 2 sin (2x) – 1. TC Calculate the average rate of change of f…

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Q: Find the linear approximation of f(x) = ln x at x = 1 and use it to estimate ln(1.33).

A: The linear approximation of f(x) at x=a is given by L(x)=f(a)+f'(a)(x−a) Given, f(x)=ln(x) at x=1.…

Q: The derivative of the function 3+ln x is: y = cos(In x)

A: y=3+ln(x)cos(ln(x)Differentiate w.r.t.xApply the Quotient Rule:  fg'=f…

Q: Find the total derivative of V with respect to X

A: As the differentiation is with respect to x, x becomes the variable. y and z becomes constants

Q: Consider the derivative function h (x) = -sec? (x) + 9. What is the general antiderivative of h?

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Q: dy Use implicit differentiation to find dx e + x² – y² = 10

A: Given implicit function is exy+x2-y2=10.

Q: The derivative of: y =0( sin (In (0))) in (In (0))) + cos(ln (0)) is is:

A: Given function is rewriting in other variables as: y=x(sin(ln(x)))+cos(ln(x)) Product rule of…

Q: What is the derivative of * In(xy) = e"y – arccos(xy)?

A: We differentiate given equation

Q: Find the second derivative of g(x) = sinh(x) cosh(x)

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Q: Find the derivative of z=(ln5)e^x dz/dx=

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Q: find the average rate of change of the function overthe given interval or intervals. g(t) = 2 + cos…

A: Given, g(t)=2+cost, -π,π The average rate of change of function f(x) over an interval [a,b] is,…

Q: Find the indic ed derivative. dz x2 for z = x2 dx 1 - x - 2x2 dz dx

A: Formulas:

Q: Find the second derivative 1. y = In(x^3+1) 2. y = Arctan (e^x)

A: Given: y=lnx3+1, y=tan-1ex

Q: Integrate ∫(ln(x))/x

A: We need to integrate the given logarithm function:

Q: Let f(x, y) = cos(x+y²). Compute the rate of change in the direction of fastest increase for f at…

A: Find gradient and its magnitude then solve

Q: Find the rate of change of y with respect to x at the indicated value of x. sin(x) y = 5 - cos(x) X…

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Q: Determine the instantaneous rate of change of the function f(x) = [x² si sin (2¹) 1+√t x3 At x=1 dt

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Q: cos(x² + y²) = y² sinx² y² = ln(x² + y²)

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Q: Find the total differential.

A: w=3x+y4z-9y

Q: Find the linear approximation of f(x)=ln x at x=1 and use it to estimate ln(1.24). L(x)=?…

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Q: Find dy/dx by implicit differentiation. x7 + y4 = 5

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Q: 1. Find the derivative of the function:F(x)= [ (Vt arctan t)dt .

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Q: Find  dy dx  by implicit differentiation. x2y3 − 12xy2 = 4

A: The given expression is, x2y3-12xy2=4 Differentiate on both sides with respect to x of the…

Q: (a) Find the first derivative of the function y= la) with respect to x by using the chain rule.

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Q: Find the differential of the function y=lntan2x at x=π/8.

A: Given: ,at x=π/8 Differentials are infinitely small quantities. Use the term for differentials as…

Q: find the derivative of y with respect to x, t, orθ, as appropriate. y = ln (cos2 θ)

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Q: Find the linear approximation of f(x)=lnx at x=1and use it to estimate ln⁡(1.3) L(x)=? ln 1.3≈?

A: First find the linear equation at x=1  then find the approximate values

Q: Using logarithmic differentiation, Show that y (x)= xa2" e®has no stationary points other than x =…

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Q: find the derivative of y with respect to x, t, or θ, as appropriate. y = ln (ln x)

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Q: 1. Find the derivative of g(x): Int dt 1+t Cos

A: This will solve by Leibnitz theorem

Q: Find the derivative of the function. y = e* In x

A: Find the derivative use product rule

Q: Show that the derivative of f(x) =xln(x)−x is ln(x). Use this to compute∫ln(x)dx.

A: Sum/difference rule of differentiation,

Q: Find dy/dx using logarithmic differentiation. y = xx − 8,    x > 0

A: