68. Use the definition of the derivative and the definition of the number e to prove that f (x) = e* is its own derivative.

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

A: Given,

Q: f(x) = (x + 2)2/3 -

A: First we have to find f'(x) . And to find extrema by first derivative test we have to find all roots…

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: Use the alternative form of the derivative to find the derivative at x = c (if it exists). (If the…

A: Here the given function  hx=x+2, c=-2 The function can be written as hx=x+2,x≥-2-x-2,x≤-2

Q: 2) Use the limit definition to find the derivative of the given function. y = 3r2 – 2x

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

Q: Use the graph of the first derivative function  f'(x) to find the interval  on which function is…

A: Given: the graph of the first derivative function  f'(x).

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

A: To find the derivative of the function fx=lnx+1

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: Use the Quotient Rule to show that the Powers of x Rule applies to negative integer powers. That is,…

A:

Q: 5. By calculating a derivative, determine whether the following statement is true (or false): et et…

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Q: 2 The derivative of (g(x))¯ is equal to (g '(x))¯. Select one: True False

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

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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: What is the derivative of the function with respect to x of (x + 1)³ – æ° -

A: Given : Function x+13 - x3   To find : Derivative of the given function with respect to x?

Q: (18) Find the derivative of the functio 2²+5 3 Y=2²-5 Nor me it to Vitou inst

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Q: 26 Find the 155th derivative of (1 + x°)" at x = 0.

A: see 2nd step.

Q: 2) NOT use the short cut method). Use the result to find the slope of the tangent line at the point…

A: Topic:- Derivative

Q: 7 Let f(x) x – 9 * Then according to the definition of derivative

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Q: g(u) u +1

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Q: Suppose 8 (x) =sec='(x<). what is the derivative of g with respect to x?

A: We can find the answer as below.

Q: Let f (x) = 7 cos(Tx). Find the twentieth derivative of f (x).

A:

Q: 84. Use the definition of the derivative to directly prove thedifferentiation rules for constant and…

A: The definition of derivative states that: f'x=limh→0fx+h-fxh Let the function be fx=c, where c is a…

Q: Prove, using the definition of derivative, that if f(x) = cos x then f ' (x) = - sin x

A: we have to prove by using the definition of derivative that if f(x)=cosx then f'(x)=-sinx   as we…

Q: b) Use the definition of the derivative to compute f'(x) for 1)f(x) = 2v-5x + 9, 2)f(x) = x sin 3x…

A:

Q: 5) Explain why the definition of a numerical derivative does not exist at x = 2 in the graph shown.…

A: The objective is to explain the reason definition of numerical derivative doesn't exist at x=2 in…

Q: 1. Evaluate the derivative of f (x)= sec(In(x²)). DO NOT SIMPLIFY.

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Q: 6. Use the definition of derivative to find f'(x) if f(x) 3D х2 — Зх — 4

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Q: 5. What is the definition of the derivative of a function f(x) ?

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Q: Find the derivative of the function. g(x) = 9 + In(x)

A: We have to find the derivative of : g(x)=ex29+ln(x)

Q: 3/ By using the definition of derivative, proof that tan x = sec? x. %3D dx

A: By definition of derivative , f'(x) = d(f(x)dx= limh→0f(x+h) - f(x)h L Hosital's Rule :When limit is…

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: Compute the derivative of the function f (x) = tan x2 +1

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Q: Find the derivative of the function. x + 4 f(x) = In X – 4 | f'(x) =

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Q: 2. Using the definition of derivative, prove the following: (a) f)=- if fo)-V

A: Find the attachment

Q: Give three different notations for the derivative of ƒ with respectto x

A: Definition used -     Derivative of a function -       It is the rate at which the y value of…

Q: The derivative of x at X = 1 is to I Frue or false 2

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Q: II) By using the limit definition of derivative, prove the identity J dzf(2)8' (2) = - / da f…

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Q: What is the derivative of the exponential function a x, a &gt;0 and a ≠ 1? What is the geometric…

A: The derivative of the exponential function ax is as follows.

Q: Find f (x) of f(x)= by using the definition of the x- derivative.

A: Definition of Derivative is given as

Q: 13 Find the secord derivative cy") y= x°+3x - when

A: see below the answer submit the q12 with what to do with the question

Q: 1 Use the limit definition of the derivative to find f'(x) if f (x)= 10x

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Q: What is the derivative of the function B(x)=3/(x+2) at x=a?

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Q: 19. Determine the derivative of f(x) = e* coshx. O a. e2x O b. ex O c. e4x O d. e5x

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Q: B) Use the definition of differentiation to find the derivative of the function f(x) = 4- Vx + 3. Is…

A: Consider the function: fx=4-x+3 The definition of derivatives is given by the formulas,

Q: Find the derivative of f with respect to x of f(x) = In x'

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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: Use the definition of derivative to compute f '(0). e-1/x if x * 0 if x - 0 f(x) f '(0) -

A: The definition of the derivative is given by: f'(x)=limh→0f(x+h)-f(x)h⇒f'(0)=limh→0f(h)-f(0)h  ...…