Question

Using the limit definition, how do you find the derivative of `f(x) = -5x^2+8x+2`?

Answer 1

`f'(x)= -10x+8`

Explanation:

Using the limit definition to find the derivativeof `f(x)`
`f'(x)=lim_(h->0)(f(x+h)-f(x))/h`
so it can be expressed as following
`lim_(h->0)((-5(x+h)^2+8(x+h)+2)-(-5x^2+8x+2))/h`
=`-10x+8`

Answer 2

Use the formula `f'(x) = lim_(h-> 0) (f(x + h) - f(x))/h`

Explanation:

`f'(x) = lim_(h->0) (-5(x + h)^2 + 8(x + h) + 2 -(-5x^2 + 8x + 2))/h`

`f'(x) = lim_(h->0)(-5(x^2 + 2xh + h^2) + 8x + 8h + 2 + 5x^2 - 8x - 2)/h`

`f'(x)= lim_(h->0) (-5x^2 - 10xh - 5h^2 + 8x + 8h + 2 + 5x^2 - 8x - 2)/h`

`f'(x) = lim_(h->0) (-10xh - 5h^2+ 8h)/h`

`f'(x) = lim_(h->0) (cancel(h)(-10x - 5h + 8))/cancel(h)`

`f'(x) = -10x - 5(0) + 8`

`f'(x) = -10x + 8`

`:.` The derivative is `dy/dx = -10x + 8`.

Hopefully this helps!