Question

How do I find the derivative of `x^2 + 7x -4` using first principles?

Answer 1

First Principles `->` Difference Quotient

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

`f(x)=x^2+7x-4`

`f(x+h)=(x+h)^2+7(x+h)-4`

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

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

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

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

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

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

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

`f'(x)=2x+(0)+7`

`f'(x)=2x+7`