Question

Using the limit definition, how do you differentiate `f(x) =4x^2`?

Answer 1

`8x`.

Explanation:

The Derivative of a given function `f` w.r.t. `x` is, denoted by `f'(x)`

& is defined by,

`f'(x)=lim_(trarrx) {(f(t)-f(x))/(t-x)}`

`f(x)=4x^2 rArr f(t)=4t^2`

Therefore, `f'(x)=lim_(trarrx) {(4t^2-4x^2)/(t-x)}`

`=4{lim_(trarrx) (t^2-x^2)/(t-x)}`

`=4[lim_(trarrx) {(t+x)(cancel(t-x))}/((cancel(t-x))]]`

`=4{lim_(trarrx) (t+x)}`

`=4(x+x)=8x`.

Enjoy Maths!