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

Find the derivative of the following functions from first principles:

1.	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
Answer» (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))`