Question

What is the derivative of ( 2w^2+1 ) ?

Answer 1

If you know about the power rule,

d/dw (2w² + 1) = 2 d/dw (w²) = 2 • 2w = 4w

If you have to use the limit definition of the derivative, let f(w) = 2w² + 1. Then

`f'(w)=\displaystyle\lim_(h\to0)\frac{f(w+h)-f(w)}h`

`f'(w)=\displaystyle\lim_(h\to0)\frac{(2(w+h)^2+1)-(2w^2+1)}h`

`f'(w)=\displaystyle\lim_(h\to0)\frac{2(w^2+2wh+h^2)-2w^2}h`

`f'(w)=\displaystyle\lim_(h\to0)\frac{4wh+2h^2}h`

`f'(w)=\displaystyle\lim_(h\to0)(4w+2h)=\boxed{4w}`