Question 555917
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Re-write the function using rational exponents:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ 4x^3\ -\ 5x\ +\ 2x^{\small{\frac{1}{2}}}]


Take the derivative of the function using the sum rule:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{d}{dx}(u\ +\ v)\ =\ \frac{du}{dx}\ +\ \frac{dv}{dx}]


And the power rule


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{d}{dx}\left(ax^n\right)\ =\ nax^{n\,-\,1}]


So


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f'(x)\ =\ 12x^2\ -\ 5\ +\ x^{\small{-\frac{1}{2}}}\ =\ 12x^2\ -\ 5\ +\ \frac{1}{\sqrt{x}}]


Now simply evaluate


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f'(4)]


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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