I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 3.3 : Differentiation Formulas
6. Find the derivative of \(f\left( x \right) = 10\,\sqrt[5]{{{x^3}}} - \sqrt {{x^7}} + 6\,\sqrt[3]{{{x^8}}} - 3\) .
Show SolutionThere isn’t much to do here other than take the derivative using the rules we discussed in this section.
Remember that you’ll need to convert the roots to fractional exponents before you start taking the derivative. Here is the rewritten function.
\[f\left( x \right) = 10\,{\left( {{x^3}} \right)^{\frac{1}{5}}} - {\left( {{x^7}} \right)^{\frac{1}{2}}} + 6{\left( {{x^8}} \right)^{\frac{1}{3}}} - 3 = 10\,{x^{\frac{3}{5}}} - {x^{\frac{7}{2}}} + 6{x^{\frac{8}{3}}} - 3\]The derivative is,
\[f'\left( x \right) = 10\,\left( {\frac{3}{5}} \right){x^{ - \,\,\frac{2}{5}}} - \frac{7}{2}{x^{\frac{5}{2}}} + 6\left( {\frac{8}{3}} \right){x^{\frac{5}{3}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{6{x^{ - \,\,\frac{2}{5}}} - \frac{7}{2}{x^{\frac{5}{2}}} + 16{x^{\frac{5}{3}}}}}\]