Question 200331
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There isn't a 'right' area to put this one.  This is an elementary calculus problem.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x) = 6x^{\frac{3}{5}} + 5\sqrt{x} - \frac{6}{3\sqrt{x}}]


Change the radicals to fractional exponents:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x) = 6x^{\frac{3}{5}} + 5x^{\frac{1}{2}} - \frac{6}{3x^{\frac{1}{2}}}]


Simplify the third term by removing the common factor of 3 from numerator and denominator and moving the variable to the numerator:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x) = 6x^{\frac{3}{5}} + 5x^{\frac{1}{2}} - 2x^{-\frac{1}{2}}]


Now, since the derivative of the sum is the sum of the derivatives, you can apply the Power Rule to each of the terms separately.


The Power Rule:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x) = ax^n \ \ \Rightarrow\ \ f'(x) = nax^{n-1}]


So:



*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f'(x) = \frac{18}{5}x^{-\frac{2}{5}} + \frac{5}{2}x^{-\frac{1}{2}} - x^{-\frac{3}{2}}] 


Which is already the proper order of terms because *[tex \Large -0.4 > -0.5 > -1.5]


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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