Find the derivative of the following functions from first principles: (i) ` x` (ii) `(-x)^(-1)` (iii) `s in (x + 1)` (iv) `cos(x-pi/8)`Find derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s a

Explanation

(i) Let `f(x)=-x`
From first principle.
`f(x)=underset(hrarr0)"lim"(f(x+h)-f(x))/(h)`
`=underset(hrarr0)"lim"(-(x+h)-(-x))/(h)`
`=underset(hrarr0)"lim"(-h)/(h)=underset(hrarr0)"lim"(-1)=-1`
(ii) Let `f(x)=(-x)^(-1)=(1)/(-x)=-(1)/(x)`
From first principle,
`f(x)=underset(hrarr0)"lim"(f(x+h)-f(x))/(h)`
`=underset(hrarr0)"lim"((-(1)/(x+h)-(-(1)/(x)))/(h)`
`=underset(hrarr0)"lim"(-(1)/(x+h)+(1)/(x))/(h)=underset(hrarr0)"lim"(-x+(x+h))/(hx(x+h))`
`=underset(hrarr0)"lim"(h)/(hx(x+h))=underset(hrarr0)"lim"(1)/(x(x+h))`
`=(1)/(x(x+0))=(1)/(x^(2))`
(iii) Let `f(x)=sin (x+1)`
From first principle,
`f(x)=underset(hrarr0)"lim"(sin(x+h+1)-sin(x+1))/(h)`
`=underset(hrarr0)"lim"(2cos(x+1+(h)/(2))sin(h)/(2))/(h)`
`=underset(hrarr0)"lim"(cos(x+1+(h)/(2))sin(h)/(2))/((h)/(2))`
`=cos(X+1+0).1=cos(x+1)`
(iv) Let `f(x)=cos(x-(pi)/(8))`
From first principle,
`f(x)=underset(hrarr0)"lim"(f(x+h)-f(x))/(h)`
`=underset(hrarr0)"lim"(cos(x+h-(pi)/(8))-cos(x-(pi)/(8)))/(h)`
`=underset(hrarr0)"lim"(-2sin(x-(pi)/(8)+(h)/(2)).sin(h)/(2))/(h)`
`=underset(hrarr0)"lim"(-sin(x-(pi)/(8)+(h)/(2)).sin(h)/(2))/((h)/(2))`
`=-sin(x-(pi)/(8)+0).1`
`=-sin(x-(pi)/(8))`