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

1 Answer
VNVDVI
Apr 18, 2018

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)

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