Question

Prove, using the definition of the derivative, that`(a f(x))'=a f'(x)` (where `a` is constant with respect to `x`)?

Answer 1

see below

Explanation:

By definition of the derivative `f'(x)=lim_(h rarr 0) ( f(x+h)-f(x) ) / h`
So we want to apply this definition to `af(x)`

`(af(x))'=lim_(h rarr 0) ( {af(x+h) } - {af(x) } ) / h`
`:. (af(x))'=lim_(h rarr 0) (a)( {f(x+h) } - {f(x) } ) / h`
`:. (af(x))'=(a)lim_(h rarr 0) ( {f(x+h) } - {f(x) } ) / h` (`a` is a constant)
`:. (af(x))'=(a)f'(x)` (by definition of `f'(x)`)

QED