1. Prove, using the definition of derivative, that if f(x) and g(x) are differentiable, then (10)- "2) • g(z) –S(2) - d(z) g²(x) (x) \ f'(x) · g(x) – f(x) · g'(x)

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

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Q: 5. Why does it make graphical sense that the derivative of aconstant is zero? That the derivative of…

A: Derivative of a function is a slope of function on given point, or derivative is a rate of change of…

Q: 1. Find the equation of the tangent line of f(x) at the given point. f (x) = 6x – x2, (1,5)…

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Q: 6. Find the tangent line to the function f(x) = 2x cos"(x) (0, 1).

A: To find the slope we need…

Q: Q-2) Prove that converting the Higher-Order derivatives to the finite difference formula would be:…

A: Hi! Thank you for the question, As per the honor code, we are allowed to answer one question at a…

Q: Show both the Newton and Leibniz notation for finding the derivative of y = f(x) = 4x5.

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Q: 8. Suppose the derivative of the function y = f(x) is y' = (x – 1)*(x – 2)(x – 4). At what points,…

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Q: • Show that both of the functions f(x) = (x − 1)^4 and g(x) = x^3 −3x^2 + 3x+ 2 have stationary…

A: given functions fx=x-14gx=x3-3x2+3x+2 (A) to show both have stationary point at x=1 differentiate…

Q: 5a) Using the formula man = lim+)=/G2 determine the derivative/ slope of tangent for h→0 the given…

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Q: a. Find the derivative of f(x) = x2 - 5x at x. That is, find f′(x). b. Find the slope of the tangent…

A: Given: fx=x2-5x To Find: a. f'x=?b. Slope of tangent line of the given function at x=-1 and x=3

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: 10 Find the derivative f'(0) of f(x) = at x + 4 x = 0 using the laws of differentiation.

A: Derivative of the function

Q: 13. Find the derivative of y = in terms of f, g, h, f', g' and h', where f, g, and h are functions…

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Q: 2. Let f(x) = /1 – x² . Find f(8) (0), i.e., the eighth derivative of ƒ evaluated at the point 0.

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: In(5æ) Find the derivative with respect to x of y = %3D 5x y' = | %3D

A: Find the derivative

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: Develop a general rule for the nth derivative of xf(x), where f is a differentiable function of x.

A: Given: the function xfx, where f is a differentiable function of x.

Q: if f(x) = x²√In x² x . find the derivative of this function f'(x) A. f'(x) = B. f'(x) = c. f'(x) =…

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Q: 15 - What is the approximate derivative of f(x) = 5x° + 4x? if we use forward difference method,…

A: Derivative of function at any point gives the slope of tangent to the function at that point.…

Q: 3- Show that this function is differentiable for all and find its derivative (2) = f(x+ jy) = x*-…

A: The necessary and sufficient condition for a complex function f(z)=U+iV  (where U,V are the real and…

Q: 1. Suppose that the revenue (in dollars) from the sale of x units of a certain product is given by…

A: The revenue function is given by Rx = 70x-0.05x2 To evaluate: The expression of R'(x) using the…

Q: (a) Let f(x)=√√2x+1. Use definition of the derivative Equation 3.4 to compute f'(x). (No other…

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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: Why does the maximum and minimum points for y = f (x) occur among the minimum and maximum points for…

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

A: Given the first derivative is: y'=6x+1x-22

Q: _6. What is the derivative of f(x) = (5 – x²)²? а. 2(5- х2) b. 10 – 2x2 C. 4x3 – 20x d.(5 - x?)(5 -…

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Q: 1 6. Find the derivative of f(x)= 1 at the point 8, Vx+1

A: Given

Q: 2. What is the equation of the tangent line to the curve y = x? - 2x +1 at point (3,4)? а. у3 2x — 2…

A: The equation of line is y-y1=mx-x1, where m is the slope of the curve.

Q: Find the derivative of y with respect to x. 1, 1+x2 In 1- x3 y

A: If you have function f(x) in the numerator and the function g(x) in the denominator, then the…

Q: 1 Find the derivative of g(x) = Vx-1 Find f'(x) and f"(x) of f(x) = 11x³.

A: Here we have to find the derivatives of the given function.

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

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Q: Find the derivative of f(x) = e" – a3 +In(x) - In(x)

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

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Q: :) Given f(x) = |x|+x, at which point does the derivative not exist? Mathematically Question 4 (-…

A: We are given a function fx=x+x and we have to find the point at which fx is not derivable. We know…

Q: 7. Consider the function f(x) X - 3 a. Using the definition of the derivative at a point, calculate…

A: Given : f(x) = x / x - 3   (1) f' (2)  (2) tangent line when x = 2 (3) normal line when x = 2

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

A: Given function is f(z)=2z+13 We have to find the derivative of the given function at z=-i.

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

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Q: Find the derivative of the function by the limit process. f(x)=x2+x-1

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Q: 3. Find the first derivative of the function f(x)= 2x3-4x+1 using forward Difference formula if, X1…

A: Solution:

Q: In x 3. Compute the approximate derivative of f(x) = · at x = 2, using three-midpoint x² + 1 formula…

A: Given is a function and at a point at which the approximate value of first derivative of the…

Q: 1. Find the first order derivative of u = x*.

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Q: find the derivative of y with respect to x, t, or θ, as appropriate.

A: On simplification, we get

Q: 15 - What is the approximate derivative of f(x) = 5x³ + 4x² if we use f(ze+h)–f(za) forward…

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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: 1) Find the derivative of the given function with respects to 0 y = 3lan In 3

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: 3- Suppose that the second derivative of the function y = f(x) is y" = (x + 1)(x - 2). For what…

A: 1)y'' = (x+1)(x-2)2)y' = (x-1)2(x-2)(x-4)

Q: 5. Find derivative of the following composite function using the Chain Rule: f(x) = 3x2 + 8x, x(t) =…

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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. Find the derivative of f(x) = x² – 3x. Use the result to find the slope of the tangent line to…

A: Since you have posted multiple questions in this question which are not interlinked so I have solved…