The Differentials: Example 7

Problem:

Find d\left[ln{\left(y^2\right)}+ln{\left(1+\sqrt y\right)}\right].

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SOLUTION:

=d\left[ln{\left(y^2\left(1+\sqrt y\right)\right)}\right]

=d\left[ln{\left(y^2+y^\frac{5}{2}\right)}\right]

=\frac{1}{y^2+y^\frac{5}{2}}\times\left(2y+\frac{5}{2}y^\frac{3}{2}\right)dy

=\frac{y\left(2+\frac{5}{2}y^\frac{1}{2}\right)}{y^2\left(1+\sqrt y\right)}dy

=\frac{2+\frac{5}{2}y^\frac{1}{2}}{y\left(1+\sqrt y\right)}dy

=\frac{2+\frac{5}{2}\sqrt y}{y\left(1+\sqrt y\right)}dy