Calculus problem

Question

Use the limit definition of the derivative to differentiate g(x)=8x-x^2

Answer 1

Use the limit definition of the derivative to differentiate g(x)=8x-x^2

The limit definition of the derivative states that for a function f(x) its derivative equals

f'(x) lim h→0 = f(x+h)−f(x) / h

g(x)=8x-x^2

g(x+h) =8(x+h) -(x+h)²

= 8x +8h-(x² +2hx +h² )

= 8x +8h-x² -2hx-h²

g'(x) lim h→0 = 8x +8h-x² -2hx-h² - (8x-x² ) / h

= 8x +8h-x² -2hx-h² - 8x + x² / h

= 8h -2hx -h² / h

=h(8-2x -h)h

=8 -2x -h

=8-2x-0

=8-2x

Answer 2

g'(x) = lim_h→0 [(g(x+h) - g(x)) / h]

= lim_h→0 [(8(x+h) - (x+h)² - 8x + x²) / h ]

= lim_h→0 [(8x + 8h - x² - 2xh - h² - 8x + x²) / h]

= lim_h→0 [(8h - 2xh - h²) / h ]

= lim_h→0 [h(8 - 2x - h) / h ] = lim_h→0 [8 - 2x - h] = 8 - 2x