2. Let f : R → R be a differentiable function. Show that the function f', the derivative function of f, is Borel measurable.

Q: If the derivative of y = f(x) at x = 2 is equal to " (0 " then y = f(x) is not continuous at r = 2.…

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Q: 1) Given f(x, y) = x derivative of f? which of the following is Əyox the mixed second order par f

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Q: Obtain the derivative of any of the following functions: A) f(x) = 3cotg²3x B) f(x) = 6cos7(3x² – 1)…

A: We use the standard formula of derivatives to solve this problem.

Q: Find the linearization of f(x) = (1+ x)20 at x = 0. Use it to approximate (1.001)20.

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Q: Let A E R"Xn, let b e R", and let n n f (x) = xT Ar + 6Tx = Aikr;xk +> b;x;. i,k=1 i=1 Show that f :…

A: Definition: A function f:ℝn→ℝm is differentiable at x0 with derivative M if M:ℝn→ℝm is linear and…

Q: Differentiate. f(x) = In ( In (8x)) f'(x) = Compute the following derivative using the method of…

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Q: Find the derivative of the function #1) F(x) = In(2x+3)· (4x +7) tan(x) t +4 dt t2 +2 # 2) h(x)= |

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Q: Prove that y = x- 2 is differentiable and calculate y'.

A: Given: The function, y = x-2

Q: Use part I of the Fundamental Theorem of Calculus to find the derivative of f(2) = / 1 -dt 1+ t…

A: We will find f'(x) as following.

Q: Differentiate f(x) with respect to g(x) of yx=xsiny

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Q: · Use the definition of the derivative of complex function to show that f(z) is not dif- ferentiable…

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Q: Define a function f : R² → R² by f 2.у 1. Show that f is differentiable on A = E R² |x > 0, y > 0}…

A: To show that the function f defined as below is differentiable on A and compute its derivative

Q: Let g (x) = f' (x) be a continuous function. If the derivative of the continuous function g (x)…

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Q: Let F(x) be the function defined as follows: F(x)= (1+ In(t))*dt What is the value of F'(e) ?

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Q: Suppose that f(a) = f(b) = g(a) = g(b) = 0 and the second derivatives of f and g are continuous on…

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Q: What does it mean for f to be differentiable at a ?

A: we have to define that what we mean by the f is differentiable at a.

Q: Let f(x, y) = V xy. Find the derivative of f. af 1 ду 2

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Q: Suppose that F(t) is a twice-differentiable function of one variable. Define the function f(x,y) =…

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Q: 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: Given: f(a) = 0, f'(a) = 0 [f'(a) is derivative of f(x) at a] To Show: (x-a)2 divides f(x) Concept…

Q: 4. Suppose that F(t) is a differentiable function of one variable. Define the function z(x, y) = e'F…

A: z(x,y) ey F (exy )diff wrt xdzdx = ey F' ex/ye.1/yzx = ey.ex/yyF' (ex/y)and diff wrt ydzdy= ey…

Q: Suppose F(z) = | e dt. Use the Fundamental Theorem of Calculus to evaluate the derivative. 13 F'(포)…

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Q: tan-lt t +0 t f(t) = 1 t = 0.

A: Given that, f(t)=1 when t=0. Differentiate the function with respect tot t implies, f'(t)=0  when…

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: Consider the For at function y= In(x+1) . what valves" of x is y undefined ?

A: The given function is y=ln (x+1) We know that, for the function ln x, x>0

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: Let f be the function with derivative given by f' (z) = sin z +I cOs z for 0 <z < 7. On which of the…

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Q: Find a positive value, c, for x, thot satisfies the Mean valve Theorem for derivatives for…

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Q: 2. Prove that f(x) = |x + 1] 1| (−1,0) tangent line at has no

A: As per the company's guidelines, we are supposed to solve first question for you, kindly repost…

Q: Suppose that u(x) is a differentiable function such that

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Q: Suppose that F(t) is a differentiable function of one variable. Define the function z(x, y) = e"F…

A: Given that zx,y=eyFexy where F(t) is differentiable function . Prove that xzx+yzy=yz

Q: Let z = x+iy and f(2)=√√|xy|. Show that ƒ(z) satisfies the Cauchy-Riemann equations at the origin,…

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Q: 4. Show in two ways that the function f(z) = exp(z²2) is entire. What is its derivative? Ans. f'(z)…

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Q: Let f (z) = (5z +3) (i – xỳhat is the real part of the derivative f' (1+ i}

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Q: Let f(x) E R[x]. Suppose that f(a) = 0 but f' (a) + 0, where f' (x) is the derivative of f(x). Show…

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Q: -2 In(x) Consider the function f(x, y) 3y

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Q: Find the derivative of the following function F(x) = (2t – 1) dt using the Fundamental Theorem of…

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Q: Find the derivative of the following function (2t – 1)* dt F(x) = using the Fundamental Theorem of…

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Q: Suppose that f(a) = f(b) = 0 and the second derivatives of f exist on the closed interval [a, b].…

A: Prove that ∫abx-ax-bf"xdx=2∫abfxdx.

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

A: Follow the procedure given below.

Q: Compute the right-hand and left-hand derivatives as limits to show that the functions are not…

A: y=x,  y=1x  &amp;  P1, 1. Left hand derivative is,…

Q: 3. Suppose that z = f(x, y) is a differentiable function with respect to variables !! x and y and is…

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Q: Suppose that f(x)=xu(x), where u(x) is a differentiable function at x=2 such that u(2)=3, u'(2)=1.…

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Q: Suppose F(z) = e dt. Use the Fundamental Theorem of Calculus to evaluate the derivative. F'(z) = dt…

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Q: The general anti-derivative of the function f (x) - In(x) is given by x In( x)+ C Inx)-x+ C x In(x)…

A: The integral (anti derivative) of fx=lnx can be evaluated by using integration by parts. The…

Q: Show whether the following function is continuously differentiable: f(x,y)=x2y2

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Q: f[in (w) ² ] ^ 4 3 dw

A: ∫lnω24ωdω=∫2lnω4ωdω=∫16ln4ωωdω=16∫ln4ωωdω Substitute u=lnω⇒dudω=1ω⇒dω=ωdu; 16∫ln4ωωdω=16∫u4du…

Q: Use the First Part of the Fundamental Theorem of Calculus to find the derivative of the following…

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Q: Let f(x) = In(1 + 7 x). Find flS(0), that is the third derivative of f evaluated at 0. f(3 !3D

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Q: 34. Show that f(x) = Jã is not differentiable at x = 0.

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