Question

F(x) = 7x^2 + 4, find and simplify using the derivative function f(a+h) -f(a)/ (h) ?

Answer 1

`(dF)/(dx) = 14x`

Explanation:

I'm not 100% sure what you are asking but I'm going to take a punt that you want to find the derivative of `F(x)` using the limit definition? You have left the limit out of your 'derivative function' but this is what I think you meant.

`(df)/(da) = lim_(h->0) (f(a+h) - f(a))/h`

With `F(x) = 7x^2 + 4` we have

`(dF)/(dx) = lim_(h->0) (7(x+h)^2 + 4 - (7x^2+4))/h`

Expanding out the squared bracket some terms helpfully cancel out:

`=lim_(h->0) (color(red)(7x^2) + 14xh + 7h^2 + color(blue)(4) color(red)(- 7x^2) color(blue)(- 4))/h`

`=lim_(hrarr0) 14x + 7h = 14x`

`therefore (dF)/(dx) = 14x`

Pleasingly this is what we would expect had we differentiated normally using the power rule.

Answer 2

We use the expression `f'(x) = lim_(h->0) (f(x + h) - f(x))/h`.

`f'(x) =lim_(h->0) (7(x + h)^2 + 4 - (7x^2 + 4))/h`

`f'(x) = lim_(h->0) (7x^2 + 14xh + h^2 + 4 - 7x^2 - 4)/h`

`f'(x) = lim_(h->0) (14xh + h^2)/h`

`f'(x) = lim_(h->0) (h(14x + h))/h`

`f'(x) = lim_(h->0) 14x + h`

`f'(x) = 14x`

This matches what we would obtain using the power rule, where `f'(x) = 7(2)x^(2 - 1) = 14x^1 = 14x`.

Hopefully this helps!