Question

How do i derive (x^-4) using the limit definition [f(x+h) - f(x) / h]

Answer 1

`f'(x)=\lim\limits_(h\to0)(f(x-h)-f(x))/(h)\\\\f(x)=x^(-4)=(1)/(x^4)\\\\f'(x)=\lim\limits_(h\to0)((x+h)^(-4)-x^(-4))/(h)=\lim\limits_(h\to0)((1)/((x+h)^4)-(1)/(x^4))/(h)=\lim\limits_(h\to0)((x^4-(x+h)^4)/(x^4(x+h)^4))/(h)\\\\=\lim\limits_(h\to0)\left[(x^4-x^4-4x^3h-6x^2h^2-4xh^3-h^4)/(x^4(x+h)^4)\cdot(1)/(h)\right]=\lim\limits_(h\to0)(-4x^3h-6x^2h^2-4xh^3-h^4)/(hx^4(x+h)^4)`


`=\lim\limits_(h\to0)(h(-4x^3-6x^2h-4xh^2-h^3))/(hx^4(x+h)^4)=\lim\limits_(h\to0)(-4x^3-6x^2h-4xh^2-h^3)/(x^4(x+h)^4)\\\\=(-4x^3-6x^2\cdot0-4x\cdot0^2-0^3)/(x^4(x+0)^4)=(-4x^3)/(x^4\cdot x^4)=(-4)/(x^5)\\\\\\D_f=D_(f')=\mathbb{R}\ \backslash\ \{0\}`