5. Use the concept of limits to find derivative of f(z) = (2z + 1)3 at z = -i.

Q: If ƒ(x) = x^n, n>=1, show from the definition of the derivative that ƒ′(0) = 0.

A: Given,

Q: 2. Find the derivative of a function at a certain number using the formula. f(x) – f(xo) f'(x) = lim…

A:

Q: Determine the derivative of f(x) = 3 x2 In x with respect to x. f '(x) :

A: We differentiate both sides with respect to x and apply the rule d/dx(uv)=ud/dx(v)+vd/dx(u)

Q: 2. Use the definition = f'(x) = lim f(w)-f(x) to find the derivative of y = tan x. dx w-x

A: Let's take; w - x = h w = x + h As w approaches x; w-x = h →0   f'(x) = limh→0  [f(x+h) - f(x)]/(h)…

Q: Use the limit definition of the derivative to prove the following special case of the Product Rule:

A: The derivative of the function is the rate of change of the value of the function when the value of…

Q: Compute dy/dx using the limit definition of the first derivative for f(x) = 3x2

A: Given function is f(x)=3x2. we have to find the derivative using limit definition .

Q: 4. Consider the function f(x) = Vx – I (a) Use the limit definition of derivative to find f'(x)

A:

Q: -10 For f(x) = 2x-3 show that f'(x) = e using the limit definition of the derivative. ind the…

A: (a) f(x)=52x-3 f'(x)=limt→xf(t)-f(x)t-x        =limt→x52t-3-52x-3t-x…

Q: Let f(x) = x^2 - 4x + 1. What is the equation of the tangent line to f(x) at x = 2? Use the limit…

A: Please see the below picture for detailed solution.

Q: Determine the derivative of f(x) = 3 x² In x with respect to x. f'(x) = |

A: We have to find f'(x)

Q: Find the derivative of f(x) = 1 at x = 3

A: The given function is: fx=1x Using the formula of derivative i.e., ddxxn=nxn-1

Q: Show that r sin (1/x) fails to have a derivative at r = 0 and even the one-sided limits (lim,-0+ (f…

A: We will prove the given statement.

Q: 5. (a) Use the limit definition of the derivative to find the derivative of f(r) = /5I+1. %3D

A:

Q: Use the limit definition of the derivative to prove the following special case of the Quotient Rule:

A: We have to prove the following derivative by using limit definition: ddxfxx=x f'x-fxx2 We know for…

Q: Determine the derivative of f(x) = ( In x )6 with respect to x. f'(x) =

A:

Q: 2. Use the definition of the derivative to find f'(x) and f"(x). a) f(x) = x? %3D

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Q: 4. Consider the function f(x) = Vx – 1 (a) Use the limit definition of derivative to find f'(x)

A: Ans:- Consider the function f(x)=x-1(a) we have to use the limit definition of derivative to find…

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: 3. Use the limit definition of the derivative to find the equation of the tangent line to the curve…

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Q: Find the derivative of the function by the limit process. h(s) = −2 (square root of x)

A:

Q: 4. Find the first order derivative with respe. to x of the function u = In/x2 + y2 + z2 %3!

A: The function is u=lnx2+y2+z2

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: 2. Determine the true error and relative true error when approximating the derivative of f (x) = -x²…

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Q: n(x)=3x?4k+1 %3D

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Q: 6. Find the first order derivative with respect to z of the function u = In/x2 + y2 +z²

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Q: Find the derivative of the function using the slope limit formula. 1 y= 3-х 1. y=(3x+1)2 2. 1 y= 1…

A: Note : As per our company guide lines, we are supposed to answer only one question. Kindly post…

Q: 11. Use the limit definition to find the derivative of f(x) = x² -2 x. 12. Find an equation of the…

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Q: 5. Find the first order derivative with respect to y of the function U = In/x² + y² + z²

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Q: 3. The derivative of f(x) = In x at x = 3 when h = 0.002

A: 3) To find the true error and relative error of the function: f(x)=ln x, x=3 & h=0.002

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

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Q: (6) Find the derivative where it exists (c) Where is flz) analytic ?

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Q: Suppose that the first derivative of y = ƒ(x) is y'= 6x(x + 1)(x - 2). At what points, if any, does…

A: The derivative of the function y=f(x) is y'=6xx+1x-2. We have to find at what points, if any, does…

Q: 4. Compute the second derivative with respect to x of 3x³ + 8 y : at x = 1.

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Q: Use the limit definition to calculate the derivative at x = a of the linear function f(x) = 11.

A: Here we have to use limit definition to find derivative

Q: 3. Express the derivative of f (x) = Vx as a limit of the difference quotient ("h → 0")

A:

Q: 2. Prove from the definition of derivative that for n EZ, d ng"-1 dr

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Q: 1. Let f(x) =1. Using the limit definition of the derivative, find f'(x).

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Q: 2. Use the definition = f'(x) = limw-x- S(w)-r(x). to find the derivative of w-x y = tan x.

A:

Q: Use the limit definition of the derivative to prove the following special case of the Product Rule:…

A:

Q: 2.Given f(x)=5Vx-1, find f'(x)using the limit definition of the derivative.

A: Given f(x)=5x-1

Q: 5. (a) Use the limit definition of the derivative to find the derivative of f(x) = V5x +1. V5yl th)-…

A: Given; f(x)=√(5x+1)

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: find the derivative of y with respect to the appropriatevariable. y = csch-1 2θ

A: Given, y=csch-12θ

Q: 6. Use the limit definition of derivative to find f'(x) for the functions below. (a) f(x) = /4r – 1…

A: limit definition of derivative is f'(x)=limh→0f(x+h)-f(x)h

Q: 3 Find the derivative with gicapeet to X If f(0 = 0-7 The derivatie with reapeet to X If -7X is…

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Q: 1. Use the definition of the derivative to find the derivative of f(x)- V-x-2

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Q: 2.) Investigate the derivate of the function f(z)=Imz by using definiton of derivative.

A: The derivative of a function is represented as the change in variable with respect to change in…

Q: suppose the derivative of f exists, and assume that f(2)=3, f'(2)=4. Let g(x)=x^2f(x) and h(x)=…

A: The objective is to find the equation of tangent lines of g(x) and h(x) at x=2

Q: 7. Use the limit definition of the derivative to find f'(x). Then find the equation of the tangent…

A: