Find the derivative of the following functions from first principle.(i) `x^3-27` (ii) `(x 1) (x 2)` (iii) `1/(x^2)` (iv) `(x+1)/(x-1)`

Explanation

(i) Let `f(x)=x^(3)-27`
From first principle, `f'(x)=underset(rarr0)(f(x+h)-f(x))/(h)`
`=underset(hrarr0)"lim"([(x+h)^(3)-27]-[x^(3)-27])/(h)`
`(x^(3)+3x^(2)h+3xh^(2)+h^(3)-27-x^(3)+27)/(h)`
`underset(hrarr0)"lim"(h(3x^(2)+3xh+h^(2)))/(h)`
`=underset(hrarr0)"lim"(3x^(2)+3xh+h^(2))=3x^(2)`
(ii) Let `f(x)=(x-1)(x-2)=x^(2)-3x+2`
From first prinicple
`f(x)=underset(hrarr0)"lim"(f(x+h)-f(x))/(h)`
`=underset(hrarr0)"lim"([(x+h)^(2)-3(x+h)+2] -(x^(2)-3x+2))/(h)`
` =underset(xrarr0)"lim"(x^(2)+2hx+h^(2)-3x-3h+2-x^(2)+3x-2)/(h)`
`=underset(hrarr0)(h[2x+h-3])/(h)`
`=underset(hrarr 0)"lim"(2x+h-3)=2x-3`.
(iii) Let `f(x)=(1)/(x^(2))`
From first principle.
`f(x)=underset(hrarr0)(f(x+h)-f(x))/(h)`
`underset(hrarr0)"lim"((1)/((x+h)^(2))-(1)/(x^(2)))/(h)`
`=underset(hrarr0)"lim"(x^( 2)-(X+h)^(2))/(hx^(2)(x+h)^(2))`
`=underset(hrarr0)"lim"(-h(2x+h))/(x^(2)(x+h)^(2))`
`=underset(hrarr0)"lim"(-2(2x+h))/(x^(2)(x+h)^(2))=-(2x)/(x^(4))=-(2)/(x^(3))`.
(iv) Let `f(x)=(x+1)/(x-1)`
`rArr f(x+h)=(x+h+1)/(x+h-1)`
`f(x)=underset(hrarr0)"lim"(f(x+h)-f(x))/(h)=underset(hrarr0)"lim"((x+h+1)/(x+h-1) -(x+1)/(x-1))/(h)`
`underset(hrarr0)"lim"((x+h+1)(x-1)-(x+1)(x+h-1))/((h(x+h-1)(x-1))`
`=underset(hrarr0)"lim"(-2h)/(h(x+h-1)(x-1))`
`=underset(hrarr0)"lim"(-2)/((x+h-1)(x-1))=-(2)/((x-1)^(2))`.