Find the derivative of the function using the definition of derivative. g(t) = 5/(√t).State the domain of the function. (interval notation.) State the domain of its derivative. (interval notation.)

Answer 1

f(t) = 5 / sqrt(t)

f(t+h) = 5 / sqrt(t+h)

f(t+h) - f(t) = 5/sqrt(t+h) - 5/sqrt(t)

= [5 sqrt(t) - 5 sqrt(t+h)] / [ sqrt(t+h)sqrt(t)]

= 5 [ sqrt(t) - sqrt(t+h)]/ sqrt( t(t+h))

Rationalizing the numerator:

= 5 [ t - (t+h)] / {sqrt(t(t+h)) * [ sqrt(t) + sqrt(t+h)]}

= (-5h) / {sqrt(t(t+h)) * [ sqrt(t) + sqrt(t+h)]}

Dividing by h:

(-5) / {sqrt(t(t+h)) * [ sqrt(t) + sqrt(t+h)]}

the limit as h tends to zer0:

(-5) / { t [ 2*Sqrt(t)]

= (-5/2) t^(-3/2)

y = 5/sqrt(t) = 5* t^(-1/2)

power rule says: (-5/2) t^(-3/2)

the domain of the original function and it's derivative is (0,infinity)