Let f(x) E R[x]. If f(a) = 0 and f'(a) = 0 [f'(a) is the derivative of f(x) at a], show that (x – a)² divides f(x).

Q: Find derivative of the function f(z) = (sin¬' (2z – 1))² with respect to z.

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Q: Find the derivative of the function f(z) = (3x +1)In(4x+3) %3D and then evaluate it at x = 0. f' (0)…

A: Given : f(x)=(3x+1)ln(4x+3)

Q: The derivative of the function y = sin-1(e* – In x), where Je – In x| < 1, is:

A: Recall ddx(sin-1(x))=11-x2 ddx(ex)=ex ddx(ln(x))=1x

Q: Find the derivative of y with respect to x. 2. y²(x + 2y) = e²

A: Given,

Q: Can you help me with this problem step by step?

A: To determine the derivative of given function.

Q: Find the derivative of function.

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Q: Find the derivative of f (x) = Ln e*+1 using the limit process. %3D

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Q: Use the derivative to find the critical point of f(x) = Ax² + Bx + C. Assume that A, B, and C are…

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Q: Find the general anti-derivative of f(x) = + Vx on a continuous interval.

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

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Q: Find the derivative of f(x)=e^c+c^x. Assume that c is a positive constant. f′(x)=

A: Consider the function  f(x)=ec+cx,       c&gt;0

Q: find the derivative of y with respect to the appropriatevariable. y = cos-1 (1/x)

A: To find dydx, rewrite the equation and then differentiate with respect to x.…

Q: Find the derivative of y with respect to x, t, or 0, as appropriate. Inx y = x7

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Q: Find the derivative at the given point. f(x)= at x = 0 %3D x+1 O 1/4 O-1

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Q: Find the derivative of the function. r(x) = In(x + 1) + 5x7ex r'(x) =

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Q: Let ƒ(x) = √2x + 1. Compute ƒ'(2),the derivative of ƒ at x = 2, and use the result to find an…

A: Consider the given function, fx=2x+1 differentiate with respect to x, f'x=ddx2x+1=122x+1·2=12x+1…

Q: Consider the derivative function h (x) = 5x³ + 3x² – 4x + 4. What is the general antiderivative of…

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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: Compute the second derivative of sin(g(x)) at x = 2, assuming that g(2) = π/4 , g'(2) = 5, and g"(2)…

A: singx

Q: Find the derivative of y with respect to x, t, or 0, as appropriate. y= In(In 9x)

A: The solution for the above question is as shown below.

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: Find the derivative of y with respect to X. y = tan 1 5x 105x(1 + 5x) 2(1 + 5x/5x J1- 5x 1+ 5x

A: It is derivative.

Q: The derivative of In x is and the derivative of a constant C is, sO X dx = In |x| +C for x> 0.

A: Given x&gt;0, The differentation of natural log x is 1/x  And differentation of constant are zero.

Q: Find the derivative of y with respect to x, t, or 0, as appropriate. 1-x y = In (x + 2)5 4x - 7 (x +…

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Q: Find the derivative of the function. F(x) = In(4 – x®) %3D f'(x) %D

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Q: Let y be a function of x and suppose y=In(x²+y²). What is the correct result for the derivative of y…

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Q: y=In(x dy -

A: Given : y = ln1xx-5

Q: Find the weak derivative of f() = |x² – x + 1/5| on the 1) interval (0,1).

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Q: The derivative of the function y = x² In(x + 1) is:

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Q: Give the derivative of f(x) arctan an(e³*) at the point where x = 0.

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Q: Suppose In (x + 2y) = xª. Use implicit differentiation to find the derivative of y with respect to…

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Q: Compute the second derivative of sin(g(x)) at x = 2, assuming that g(2) = 4, 8'(2) = 5, and g"(2) =…

A: Given: The function singx with g2=π4, g'2=5, and g''2=3, To find: The second derivative of given…

Q: Find the derivative of f(x) and solve for x sinx a. f(x) X = 6. b. f(x) e* sinx X = 0 COSX c. f(x) X…

A: Given: (1) f(x) = sin x / x  at x = π/6  (2).  f(x) = ex sinx at  x= 0 (3).  f(x) =  cos x / ex   at…

Q: Find the derivative of the function. f(x) = e In( Vx + 8 f'(x) =

A: Given, f(x) = ex ln(x+8) We have the formulas d(ex)/dx = ex                       ...........(1)…

Q: Find the derivative of the function f(x) = V (x² – 2)* at x = 1.

A: Given f(x)=(x2-2)23      ⇒f(x)=x2-2213⇒f(x)=x2-223

Q: Find the derivative of the function. 4 f(x) x In(x) f'(x) %D

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Q: Let y be a function of x and suppose y= In(y-x). What is the correct result for the derivative of y…

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Q: Find the derivative of y with respect to x, t, or 0, as appropriate. In x 20) у 3

A: Given

Q: Assuming that x> 0, use differentiation to justify the formula X dx = In |x| + C. (=| d dx = In |x|+…

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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 particular antiderivative of the derivative f'(x) = 3 -+x that satisfies the condition f…

A: Follow the procedure given below.

Q: If ƒ(x) = xn, n ≥ 1, show from the definition of the derivative that ƒ'(0) = 0.

A: Now the derivative of a function f at a point 'a' is given by,

Q: Give the derivative formula for the function. (x)2 In x -e j'(x)

A: Consider the given function.

Q: Compute the second derivative with respect to r of 3x + 8 y = at x = 1.

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Q: Suppose ln (x + 2y) = x4.Use implicit differentiation to find the derivative of y with respect to x.…

A: Differentiate the given function implicitly as follows.

Q: Find the derivative with respect to x of h(x) = e2x²+6 In(x).

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Q: Find the derivative off with respect to x of f(x) = In x⁹. The derivative off with respect to x of…

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Q: The derivative of f(z) = Log(2iz - 4+6i) at z = -3+i is

A: