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Calculating Derivative – A function with square roots – Exercise 6261

Exercise

Find the derivative of the following function:

f(x)=\frac{3}{\sqrt[3]{x}}-\frac{4}{\sqrt{x^5}}

Final Answer


f'(x)=-\frac{1}{\sqrt[3]{x^4}}+10\cdot\frac{1}{\sqrt{x^7}}

Solution

We simplify the function before differentiating:

f(x)=\frac{3}{\sqrt[3]{x}}-\frac{4}{\sqrt{x^5}}=

=3x^{-\frac{1}{3}}-4x^{-\frac{5}{2}}

Using Derivative formulas, we get the derivative:

f'(x)=3\cdot (-\frac{1}{3})x^{-\frac{4}{3}}-4\cdot (-\frac{5}{2})x^{-\frac{7}{2}}

One can simplify the derivative:

=-x^{-\frac{4}{3}}+10x^{-\frac{7}{2}}=

=-\frac{1}{\sqrt[3]{x^4}}+10\cdot\frac{1}{\sqrt{x^7}}

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