Find the derivative of the function using the definition of derivative.State the domain of the function and the domain of its derivative f(x)=5-4x+3×^2

Question

Find the derivative of the function using the definition of derivative.State the domain of the function and the domain of its derivative f(x)=5-4x+3×^2

Answer 1

Greetings! Lets solve this shall we ?

We yield the domain of the original function f(x) = 3x²-4x+5 is all real numbers or (-∞, ∞) because it is parabolic in nature. Now lets find the derivative using the definition of a derivative such that,

lim f(x+h) - f(x)

h-> 0 -------------------

h

f(x+h) = 5 - 4(x+h) + 3(x+h)² = 5 - 4x - 4h + 3x² + 6xh + 3h²

Then,

lim [5 - 4x - 4h + 3x² + 6xh + 3h²] - [5-4x+3x²]

h->0 ----------------------------------------------------------- ; Distribute the negative into the function then simplify

h

lim -4h+6xh+3h² lim h( -4 + 6x + 3h) lim

h->0 --------------------- = h->0 ------------------------ = h->0 (-4+6x+3h) = 6x-4

h h

Since (6x-4) is the derivative of the function, the domain of it is also all real numbers or (-∞, ∞) because the derivative is linear.

I hope this helped!