Question

Derivative `tan(sqrt(x))` using first principle method?

Answer 1

`f^'(x)=1/(2sqrtx)sec^2sqrtx`
We note that,
`color(red)((1)tan(A-B)=(tanA-tanB)/(1+tanAtanB)`
`color(red)((2)lim_(theta to0)tan theta/(theta)=1`

Explanation:

Here, `f(x)=tansqrtx`
For first principle method, we take
`color(blue)(f^'(x)=lim_(t tox)(f(t)-f(x))/(t-x)`
`:.f^'(x)=lim_(t tox)(tansqrtt-tansqrtx)/(t-x)`

`=lim_(t tox)(color(red)((tansqrtt-tansqrtx)/(1+tansqrtt tansqrtx)))/(t-x)xx(1+tansqrtt tansqrtx)`

`=lim_(t tox)(tan(sqrtt-sqrtx)/(t-x))xxlim_(t tox)(1+tansqrtt tansqrtx)`

`=(1+tansqrtx tansqrtx)lim_(t tox)(tan(sqrtt-sqrtx)/(sqrtt-sqrtx))1/(sqrtt+sqrtx)`

`=(1+tan^2sqrtx)lim_((sqrtt-sqrtx) to 0)(tan(sqrtt-sqrtx)/(sqrtt-sqrtx))1/(sqrtt+sqrtx)`

`=(sec^2sqrtx)(1)(1/(sqrtx+sqrtx))`

`f^'(x)=1/(2sqrtx)sec^2sqrtx`