Find the derivative of f(x)=2x^2+3 (please show all steps) using the definition of the derivative?

Answer 1

f (x) = a.x^n
f `(x) = na.x^(n-1)
f(x) = K where K is a constant
f `(x) = 0
Above rules are now used:-
f(x) = 2x² + 3
f `(x) = 4x^1 + 0
f `(x) = 4x

Answer 2

The definition of a derivative is

lim (∆x -> 0) f(x + ∆x) - f(x) / ∆x

substitute x + ∆x for x in 2*x^2 + 3 to get 2*(x + ∆x)^2 + 3.

Expand the 2*(x + ∆x)^2 term:

2* x^2 + 4*x*∆x + 2*∆x^2 + 3.

Now subtract f(x) = 2*x^2 + 3; the 2*x^2 and the 3 terms cancel out leaving

4*x*∆x + 2*∆x^2

Now divide this by ∆x to get

4*x + 2*∆x

then let ∆x -> 0 to get

4*x

which is the answer.

Answer 3

Here's how to do it using the definition of the derivative (which is what the problem asks for):

Lim[h → 0] ( f(x+h) - f(x) ) / h =

Lim[h → 0] ( (2(x+h)^2 +3) - (2x^2 +3) ) / h =

Lim[h → 0] ( 2(x+h)^2 - 2x^2 ) / h =

2 * Lim[h → 0] ( (x^2 + 2hx + h^2) - x^2 ) / h =

2 * Lim[h → 0] (2hx + h^2) / h =

2 * Lim[h → 0] (2x + h) =

2(2x+0) =

4x