1. Use the definition of the derivative as a limit to find the derivative of the function f(x)=−2x^3+3x^2+2x−10.

Question

Please explain in steps how to do this correctly and clearly.

Answer 1

the derivative = -6x^2 +6x +2

the derivative = the limit of (f(x+h) -f(x))/h as h approaches zero

f(x+h) = -2(x+h)^3 + 3(x+h)^2 +2(x+h) -10

= -2(x^3 +3hx^2 + 3xh^2 + h^3) + 3(x^2 +2xh + h^2) + 2x +2h -10

=2x^3 +6hx^2+ 6xh^2 + 2h^3 +3x^2 + 6xh + 3h^2 + 2x + 2h -10

= 2x^3 + (6h +3)x^2 + (6h +6h^2+2)x + 2h^3 + 3h^2 +2h- 10

f(x) = -2x^3 + 3x^2 +2x -10

f(x+h) - f(x) = -2x^3 + 2x^3 + (6h+3 -3)x^2 + (6h +6h^2 +2 -2)x + 2h^3 +3h^2 +2h -10+10

= -6hx^2 + 6h(1+h)x + 2h^3 +3h^2 +2h

[f(x+h) - f(x)]/h =[-6x^2 + 6(1+h)x + 2h^2 +3h + 2

substitue 0 for h to find the limit as h approaches zero

lim as h approaches 0 of 6x^2 + 6(1+h)x + 2h^2 +3h +2

= -6x^2 + 6x + 2 = f'(x)

as h approaches zero = f'(x) approaches -6x^2 +6x +2 = the derivative of f(x)