Answer
$y=\displaystyle \frac{3}{2}x-\frac{3}{2}$
Work Step by Step
Use:
Theorem 5.2 :properties of ln(x),
Theorem 5.3.1:$\ \ \ (\displaystyle \ln x)^{\prime}=\frac{1}{x}$
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$y=f(x)=\ln x^{3/2}$=... property (3) of ln...$=\displaystyle \frac{3}{2}\ln x$
$f^{\prime}(x)=\displaystyle \frac{3}{2}\cdot\frac{1}{x}=\frac{3}{2x}$
The slope of the tangent at (1,0) is $f^{\prime}(1)=\displaystyle \frac{3}{2}$
Point-slope equation of the tangent line:
$y-y_{0}=m(x-x_{0})$
$y-0=\displaystyle \frac{3}{2}(x-1)$
$y=\displaystyle \frac{3}{2}x-\frac{3}{2}$
For (b) and (c), see image attached.
