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: 2h lim h→0 h(/2x+2h+/2x)

A: Given, limh→0 2hh2x+2h+2x The definition of derivative is given by fx=limh→0 fx+h-fxh

Q: The function f (x) called invariant under differentiation operator function (IUDO. function) if:…

A: The given functions are ux=-2coshxvx=-sinhx+coshx Now firstly we will find the derivatives of u(x)…

Q: f(x) = 2+6x -x'use the limit definition of derivative t

A:

Q: Consider a quadratic function f(x) = ax?+ bx + c, where a, b, CER and a # 0. She hat the tangent…

A: f(x)=ax2+bx+climit definition of derivative states that :f'(x) =limh→0f(x+h)-f(x)hf'(x) =limh→0…

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

A:

Q: Solve the differential equation: (x + y)dx + (x + y²)dy = 0 By using the exact equation.

A: Firstly we comparing the differential equation with P dx+Q dy =0 then we check it is exact or not…

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

A:

Q: 3. Assume that f and g are entire. Use the definition of the derivative: f(z)-f(zo) 2-20 f'(20) =…

A:

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

A:

Q: If L:V→ V is a linear operator and x E Ker(L), then L(v + x) = L(v), V vEV. Select one: True 26…

A:

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: