Find the derivative of the following functions using first principle of derivative.
`tan(x + 1)`
(ii) `cot (x^(2) + 1)`

Explanation

(i) Let `f(x) = tan (x + 1)` and `f(x + h) = tan (x + h + 1)`
`f'(x) = underset(h to 0)(lim) (tan (x + h + 1) - tan (x + 1))/(h) = underset(h to 0)(lim) ((sin(x + h+1))/(cos(x + h + 1)) - (sin)(x+1)/(cos(x + 1)))/(h)`
`= underset(h to 0)(lim) (sin(x + h + 1) cos (x + 1) - sin (x + 1) cos (x + h + 1))/(h cos (X + h + 1)cos (x + 1))`
`= underset(h to 0)(lim) (sin (x + h + 1 - x - 1))/(h cos (x + + 1)cos (x + 1))`
`= underset(h to 0)(lim) (sinh)/(h) underset(h to 0)(lim) (1)/(cos(x + h + 1) cos (x + 1)) = (1)/(cos^(2)(x + 1)) = sec^(2) (x + 1)`
`:.f'(tan(x + 1)) = sec^(2) (x + 1)`
(ii) Let `f(x) = cot (x^(2) + 1)`. Then `f(x + h) = cot (x + h)^(2) + 1)`
`:.f'(x) = underset(h to 0)(lim) (cot((x + h)^(2) + 1) - cot(x^(2) + 1))/(h)`
`= underset(h to 0)(lim) ((cos((x + h)^(2) + 1))/(sin((x + h)^(2) + 1)) - (cos(x^(2) + 1))/(sin(x^(2) + 1)))/(h)`
`= underset(h to 0)(lim) (sin(x^(2) + 1) cos((x +h)^(2) + 1) - sin((x + h)^(2) + 1) cos (x^(2) + 1))/(h sin ((x + h)^(2) + 1) sin(x^(2) + 1))`
`= underset(h to 0)(lim) (sin(x^(2) + 1 - (x + h)^(2) - 1))/(sin(x + h)^(2) + 1) sin (x^(2) + 1))`
`underset(h to 0)(lim) (sin (-h^(2 - 2xh))/(h sin((x + h)^(2) + 1) sin(x^(2) + 1))`
`= -1 underset(h to 0)(lim) (sin(h(h + 2x)))/(h(h + 2x)) (h + 2x) underset(h to 0)(lim) (1)/(sin((x + h)^(2) + 1) sin(x^(2) + 1))`
`= - 1 underset(h to 0)(lim) (h + 2x) underset(h to 0)(lim) (1)/(sin((x + h)^(2)+ 1) sin(x^(2) + 1))`
`= -1.2x (1)/(sin^(2) (x^(2) + 1)) = - 2x cosec^(2) (x^(2) + 1)`
`:.f(cot(x^(2) + 1)) = -2x cosec^(2) (x^(2) + 1)`