5. Let a E R. Let f be a function which is differentiable at a. Assume that f is always positive. We define a new function g by g() = Prove that g is differentiable f(x)' at a and that -f'(a) (F(a)) " Write a proof directly from the definition of derivative as a limit without using any g' (a) = of the differentiation rules (such as quotient rule or chain rule).

Q: Use the LIMIT DEFINITION of derivative to find f'(x) if f(x) =; 3x+2 LUse differentiation rules only…

A: Using definition of derivative, the derivative of function f(x) is given by: f'(x)=limh→0…

Q: Suppose F(x) is a differentiable function for all real numbers x. Evaluate the following derivative.…

A: We need to find the derivative of the function F(x^4) with respect to x.

Q: +20x-22y-14-0 become when referred to the rectangul axes through the point (-2, -3), the new axes…

A:

Q: Let g(x)=(1)/(1+x). Use the limit definition of the derivative to find g'(1). Use the formula…

A:

Q: Solve using method of undetermined coefficients: y" + y' - 2y = x². %3D

A: Solution :-

Q: Let f be the function with derivative defined by f'(x) = cos(x²) on the interval -2 < x <2. How many…

A:

Q: Use the definition of the derivative (that is, f'(a) = lim sec(x) da to compute h+0

A:

Q: Given f(x) = = ƒ'(x)= 3 X " find f'(x) using the limit definition of the derivative.

A: Given , to find  by using the limit definition of the derivative.

Q: If f has a derivative at x = 2, then f is continuous at x = = 2. True False

A:

Q: Prove that v5 is irrational.

A:

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

A:

Q: A function f: R → R has a third order continuous derivative. Show that f(a) f'(a). f"(a). f"" (a) ≥…

A: Given f:ℝ→ℝ has a third order continuous derivative Case 1: Consider fx is even function f-x=fx…

Q: Let f,g: RR be differentiable everywhere. Assume that g is one-to-one function. Show that f(g(x…

A: Let f,g: R → R be differentiable everywhere. Assume that g is one-to-one function. We have to show…

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: