Question

How do you differentiate `y=2^(arcsin(sqrtt))`?

Answer 1

`dy/dt=(2^arcsinsqrttln2)/(2sqrt(t-t^2)`

Explanation:

We can use logarithmic differentiation:

`y=2^arcsin(sqrtt)`

Apply the logarithm to both sides:

`lny=ln(2^arcsinsqrtt)`

Recalling that `ln(a^b)=blna,` we get

`lny=arcsinsqrttln2`

Now, differentiate both sides with respect to `t`. This will result in implicitly differentiating the left side. Furthermore, recall that `d/dxarcsinx=1/sqrt(1-x^2)`

`1/y*dy/dt=ln2/sqrt(1-(sqrtt)^2)*d/dtsqrtt`

`1/y*dy/dt=ln2/(2sqrt(t-t^2)`

Solve for `dy/dt:`

`dy/dt=y*ln2/(2sqrt(t-t^2)`

`dy/dt=(2^arcsinsqrttln2)/(2sqrt(t-t^2)`