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Solution assignment 21 Differentiation of standard functions

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Assignment 21

Differentiate:

y=(x^3+\sqrt{x})\sqrt[5]{x^3}

Solution

This seems a complicated function and indeed, it is more difficult than the previous functions. However, recall that a root function can be written as a power function. In this case the function can be rewritten as:

y=(x^3+x^{\frac{1}{2}})x^{\frac{3}{5}}=x^{3\frac{3}{5}}+x^{\frac{1}{2}+\frac{3}{5}}=x^{3\frac{3}{5}}+x^{1\frac{1}{10}}

This is the sum of two power functions and thus the derivative is:

y'=3\frac{3}{5}x^{3\frac{3}{5}-1}+1\frac{1}{10}x^{1\frac{1}{10}-1}=3\frac{3}{5}x^{2\frac{3}{5}}+1\frac{1}{10}x^{\frac{1}{10}}

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