Question: use the first principles of a derivative to find (dy÷dx) f(x)= -x^4 +4x^4
So we need to use the formula f(x+h)- f(x) ÷ h
So far I have x^4 - h^4 -4h^3x -6h^2x^2 - 4h^3-x^4 + 4x^2 +8hx+ 4h^2÷ h is this correct and what do I do after
- x ^ 4 + 4 x ^ 4 = 3 x ^ 4
(dy÷dx) f(x)= 3 limit as h -> 0 [ ( x + h ) ^ 4 - x ^ 4 ] / h =
3 limit as h -> 0 [ ( x ^ 4 + 4 x ^ 3 h + 6 x ^ 2 h ^ 2 + 4 x h ^ 3 + h ^ 4 ) - x ^ 4 ] / h =
3 limit as h -> 0 ( 4 x ^ 3 h + 6 x ^ 2 h ^ 2 + 4 x h ^ 3 + h ^ 4 ) / h =
3 limit as h -> 0 ( 4 x ^ 3 h / h + 6 x ^ 2 h ^ 2 / h + 4 x h ^ 3 / h ^ 2 + h ^ 4 / h ) =
3 limit as h -> 0 ( 4 x ^ 3 + 6 x ^ 2 h + 4 x h ^ 2 + h ^ 3 ) =
3 [ limit as h -> 0 ( 4 x ^ 3 ) + limit as h -> 0 ( 6 x ^ 2 h ) + limit as h -> 0 ( 4 x h ^ 2 ) + limit as h -> 0 (h ^ 3 ) ]
When h -> 0 then
limit as h -> 0 ( 6 x ^ 2 h ) -> 0 , becouse 6 x ^ 2 * 0 = 0
limit as h -> 0 ( 4 x h ^ 2 ) -> 0 , becouse 4 x 0 ^ 2 = 0
limit as h -> 0 (h ^ 3 ) -> 0 , becouse 0 ^ 3 = 0
So :
3 [ limit as h -> 0 ( 4 x ^ 3 ) + limit as h -> 0 ( 6 x ^ 2 h ) + limit as h -> 0 ( 4 x h ^ 2 ) + limit as h -> 0 (h ^ 3 ) ] =
3 ( 4 x ^ 3 + 0 + 0 + 0 ) =
3 * 4 x ^ 3 =
12 x ^ 3