By choosing small values for h, estimate the instantaneous rate of change of the function f(x) = x³ with respect to x at x = 1.

Q: Find the derivative of the function.…

A: The given function is hx=x2arctan8x

Q: Find the derivative of the function y = ln(tanh x/2)

A:

Q: Find the derivative of function.

A: To find the derivative of the given function                                                y=log…

Q: Find the derivative of the function f(x) = x√ln x

A: Here given Use the formula of product rule of derivatives as

Q: Find the derivative gx) = 4 K) = cost dt

A: Problem is solved using Leibniz rule

Q: Find the derivative of function. y = log(2x)

A: A composite function is differentiated as follows: ddxfgx=f'gx·g'x

Q: Find the first order derivative with respect to y of the function u = log(x + x² – y2)

A: Using derivatives.

Q: Let y=sqrt(2x). Find the rate of change of y with respect to x at x=4.

A: Answer is 1/(2√2).

Q: Find the first to third derivative of the following functions and evaluate the rate of change with…

A:

Q: Find the derivative of the function. y = (ln x)4

A:

Q: Find the derivative of the function.

A: Given y=lnx+1x-1

Q: Estimate the true percent relative error for the first derivative of the function, y =[eln)P, at x =…

A: We can solve this using Central difference formula f'(x) = (f(x+h) -f(x-h))/(2h)   The value of…

Q: Find the derivative of the function. y = log 4 V3x +2

A: The Answer is Option A

Q: Find the nth-order derivative of the function y=(3x2−2x)lnx, x>0.

A: Use the product rule

Q: Find the derivative of the function.

A: Given: f(x)=log2x2x-1

Q: Find the derivative of the function f(x) = ln(sinh x)

A:

Q: Calculate the 4th derivative of the function y=x2/x−1 at x=2.

A:

Q: Find the derivative of the function y = ln(9 – e-"). f'(x) =

A: Given that y=ln(9-e-x) To find : Differentiation  y'=f'(x)=ddx(ln(9-e-x))=19-e-xddx(9-e-x)=e-x9-e-x

Q: Compute backward difference approximations for the first derivative of the function, y = log(x? +…

A: Given is the function y=log(x2+1)2   .  It is needed to compute backward difference approximation…

Q: find the derivative of y with respect to the givenindependent variable. y = log2 r . log4r

A: Recall the properties of logarithms:

Q: Determine the 3rd derivative of the function y=x3/x+2 at x=−1.

A:

Q: Compute the derivative: x4 – 4 d - at x = 2 dx x2 – 5 - The derivative is:

A:

Q: 5. First compute the derivative of f(x)= Vĩ and then use it to: a. Find the equation of the tangent…

A: Given equation of curve is fx=x. Power rule of differentiation states that: ddxxn=nxn-1…

Q: (d) y = xnz²

A:

Q: find the derivative of y with respect to the givenindependent variable. y = 3 log8 (log2 t)

A:

Q: Find the derivative of the function. y= log 4- x

A:

Q: Suppose y = f(x) is a curve that always lies above the x-axis and never has a horizontal tangent,…

A:

Q: Find the (a) approximate error and (b) relative approximate error of the following: The derivative…

A: Note:- For this problem, you don't mention any particular method and anything about h. So, I'll try…

Q: 2. Given that y= -- Use a limit to find the instantaneous rate of change of y with respect to x at…

A:

Q: The derivative f'(a) represents the instantaneous rate of change of y = f (x) with respect to x when…

A: Given that we need to check  behaviour of  the statement " The derivative f'(a) represents the…

Q: Use Richardson extrapolation to estimate the first derivative of Y = sin x at * = π/4 using step…

A: Given the function  y=sinx

Q: The derivative of xª at x = e is equal to .Soloot ono

A:

Q: Find the derivative of the function: f(x)=ln[x^2÷(x+1)^3

A: Formula:

Q: 2. If f (x) = xV x + 3, find the derivative of this function and the slope of the tangent line at…

A: Product rule of differentiation: (fg)'=fg'+gf' The slope of a tangent line at (a,b) is given by:…

Q: Find the function with the derivative f'(x) 3x = e whose graph passes through the point P| 0,

A:

Q: find the derivative of y with respect to the givenindependent variable. y = log2 (8tln 2)

A: We first expand the equation using the addition property of log and the change of base property.

Q: f y = (sin x)x b

A:

Q: Find the derivative of the function. f(x) = ln x/(x2 + 1)

A:

Q: Find the derivative of the function.

A:

Q: 9. Consider the interval [0, 2n] for the function y sec x. State an x- value at which the…

A: Given query is to find value of x for different conditions for instantaneous rate of change.

Q: Let m(x) be the slope of the tangent line to the curve y = 2x° – 6x² + 3x at the point (x, y). Find…

A: We have to find instantaneous rate of change of m(x) at point (2,-2)

Q: Find the derivative of F(x) = t°e' ln tdt.

A:

Q: (4)The instantaneous rate of change of f(r) = e- at a = 1 is . .. (5) The derivative of f(x) = sec2…

A: (4) We have So we get the instantaneous rate of change of the function at  as .

Q: Find the first derivative or f(x) of the following functions using the techniques of integration.

A:

Q: Let m(x) be the slope of the tangent line to the curve y = 2x³-6x² + 3x at the point (x, y). Find…

A: Given: y=2x3-6x2+3x

Q: find the derivative of y with respect to the givenindependent variable. y = log3 (1 + θ ln 3)

A:

Q: Estimate the true percent relative error for the first derivative of the function, y =[e!n(x)72, at…

A: We can solve this using Central difference formula f'(x) = (f(x+h) - f(x-h))/(2h)     x 0.95 1…

Q: Find the derivative of the function f(x) below. f(x)=ln(5−e^(−4x)) f′(x)=

A: Explanation: Given that, f(x)=ln(5-e-4x) Differentiate both sides with respect to x.…

Q: find the derivative of y with respect to the givenindependent variable.

A: