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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,
\[f'\left( x \right) = 10\,\left( {\frac{3}{5}} \right){x^{ - \,\,\frac{2}{5}}} - \frac{7}{2}{x^{\frac{5}{2}}} + 6\left( {\frac{8}{3}} \right){x^{\frac{5}{3}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{6{x^{ - \,\,\frac{2}{5}}} - \frac{7}{2}{x^{\frac{5}{2}}} + 16{x^{\frac{5}{3}}}}}\]