# Math Help - differentiation w/fractions

1. ## differentiation w/fractions

Directions: Differentiate the given function doing as much of the computation as possible in your head and simplify your answer.

Problem: f(x) = square root of x^3 + 1/square root of x^3

My solution: f(x) = 3x^-1/2 + 3/2square root of x^5 = 3/2 square root of x + 3/2 square root of x^5.

2. Hello, Becky!

Differentiate the given function doing as much of the computation as possible in your head and simplify your answer.

Problem: $f(x) \;= \;\sqrt{x^3} + \frac{1}{\sqrt{x^3}}$
We have: $f(x) \;= \;x^{\frac{3}{2}} + x^{-\frac{3}{2}}$

Differentiate: $f'(x)\;=\;\frac{3}{2}x^{\frac{1}{2}} - \frac{3}{2}x^{-\frac{5}{2}}$

Factor: $f'(x) \;= \;\frac{3}{2}x^{-\frac{5}{2}}\left(x^3 - 1\right) \;= \;\frac{3(x^3 - 1)}{2x^{\frac{5}{2}}} \;= \;\frac{3(x^3 - 1)}{2\sqrt{x^5}}$