Question

How do you use the definition of a derivative to find the derivative of `f(x)=2x^2+5x`?

Answer 1

`f'(x ) = 4x+5`

Explanation:

By definition of the derivative:

`f'(x)=lim_(h rarr 0) ( f(x+h)-f(x) ) / h`

So with `f(x) = 2x^2+5x` we have;

`f'(x ) = lim_(h rarr 0) ( {2(x+h)^2+5(x+h)} -{2x^2+5x} ) / h`
`" " = lim_(h rarr 0) ( {2(x^2+2hx+h^2)+5x+5h} -{2x^2+5x} ) / h`
`" " = lim_(h rarr 0) ( {2x^2+4hx+2h^2+5x+5h} -{2x^2+5x} ) / h`
`" " = lim_(h rarr 0) ( 2x^2+4hx+2h^2+5x+5h - 2x^2-5x ) / h`
`" " = lim_(h rarr 0) ( 4hx+2h^2+5h ) / h`
`" " = lim_(h rarr 0) ( 4x+2h+5)`
`" " = 4x+5`