प्रथम सिद्धान्त से निम्नलिखित फलनों के अवकलज ज्ञात कीजिए :
(i) `x^3-27`
(ii) `(x-1)(x-2)`
(iii) `(1)/(x^2)`
(iv) `(x+1)/(x-1)`

Explanation

(i) माना `f(x)=x^3-27`
प्रथम सिद्धान्त से, `f'(x)=lim_(hto0)(f(x+h)-f(x))/(h)`
`=lim_(hto0)([(x+h)^3-27]-[x^3-27])/(h)`
`x^3+3x^2h+3xh^2+h^3`
`=lim_(hto0)(h(3x^2+3xh+h^2))/(h)`
`=lim_(hto0)(3x^2+3xh+h^2)=3x^2`
(ii) माना `f(x)=(x-1)(x-2)=x^2-3x+2`
प्रथम सिद्धान्त से,
`f'(x)=lim_(hto0)(f(x+h)-f(x))/(h)`
`=lim-(hto0)([(x+h)^2-3(x+h)+2]-(x^2-3x+2))/(h)`
`=lim_(xto0)(x^2+2hx+h^2-3x-3h+2-x^2+3x-2)/(h)`
`=lim_(hto0)(h[2x+h-3])/(h)`
`lim_(hto0)(2x+h-3)=2x-3`
(iii) माना `f(x)=(1)/(x^2)`
प्रथम सिद्धान्त से,
`f'(x)=lim_(hto0)(f(x+h)-f(x))/(h)`
`=lim_(hto0)((1)/((x+h)^2)-(1)/(x^2))/(h)`
`lim_(hto0)(x^2-(x+h)^2)/(hx^2(x+h)^2)`
`lim_(hto0)(-h(2x+h))/(hx^2(x+h)^2)`
`=lim_(hto0)(-(2x+h))/(x^2(x+h)^2)=-(2x)/(x^4)=-(2)/(x^3)`
(iv) माना `f(x)=(x+1)/(x-1)`
`rArr f(x+h)=(x+h+1)/(x+h-1)`
`f'(x)=lim_(hto0)(f(x+h)-f(x))/(h)=lim_(hto0)((x+h+1)/(x+h-1))-((x+1)/(x-1))/(h)`
`=lim_(hto0)((x+h+1)(x-1)-(x+1)(x+h-1))/(h(x+h-1)(x-1))`
`lim_(hto0)((x^2-1+hx-h)-(x^2-1+hx+h))/(h(x+h-1)(x-1))`
`=lim_(hto0) (-2h)/(h(x+h-1)(x-1))`
`=lim_(hto0)(-2)/((x+h-1)(x-1))=-(2)/((x-1)^2)`