How do I derive \( x^{-4} \) using the limit definition \(\frac{f(x+h) - f(x)}{h}\)?

f ′ ( x ) = h → 0 lim ​ h f ( x − h ) − f ( x ) ​ f ( x ) = x − 4 = x 4 1 ​ f ′ ( x ) = h → 0 lim ​ h ( x + h ) − 4 − x − 4 ​ = h → 0 lim ​ h ( x + h ) 4 1 ​ − x 4 1 ​ ​ = h → 0 lim ​ h x 4 ( x + h ) 4 x 4 − ( x + h ) 4 ​ ​ = h → 0 lim ​ [ x 4 ( x + h ) 4 x 4 − x 4 − 4 x 3 h − 6 x 2 h 2 − 4 x h 3 − h 4 ​ ⋅ h 1 ​ ] = h → 0 lim ​ h x 4 ( x + h ) 4 − 4 x 3 h − 6 x 2 h 2 − 4 x h 3 − h 4 ​
= h → 0 lim ​ h x 4 ( x + h ) 4 h ( − 4 x 3 − 6 x 2 h − 4 x h 2 − h 3 ) ​ = h → 0 lim ​ x 4 ( x + h ) 4 − 4 x 3 − 6 x 2 h − 4 x h 2 − h 3 ​ = x 4 ( x + 0 ) 4 − 4 x 3 − 6 x 2 ⋅ 0 − 4 x ⋅ 0 2 − 0 3 ​ = x 4 ⋅ x 4 − 4 x 3 ​ = x 5 − 4 ​ D f ​ = D f ′ ​ = R \ { 0 }

Answered by Anonymous | 2024-06-10

To derive x − 4 using the limit definition, we substitute the function into the derivative formula, simplify the expression using binomial expansion, and then take the limit as h → 0 . The final result is that the derivative is f ′ ( x ) = − x 5 4 ​ .
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Answered by Anonymous | 2024-12-23