Let's find the derivative of the function
[tex]\[ y = \frac{1}{\sqrt[3]{x}} + 20 \cos(x) - \frac{\sqrt[5]{x}}{\sqrt[4]{x}} \][/tex]
Step-by-step:
1. Rewrite the Function in Terms of Exponents:
To make differentiation easier, express the radicals and roots as exponents:
[tex]\[ y = x^{-\frac{1}{3}} + 20 \cos(x) - \frac{x^{\frac{1}{5}}}{x^{\frac{1}{4}}} \][/tex]
2. Simplify the Fractional Exponents:
Apply the properties of exponents to simplify the fraction:
[tex]\[ \frac{x^{\frac{1}{5}}}{x^{\frac{1}{4}}} = x^{\frac{1}{5} - \frac{1}{4}} = x^{\frac{4 - 5}{20}} = x^{-\frac{1}{20}} \][/tex]
Now, the function [tex]\( y \)[/tex] is:
[tex]\[ y = x^{-\frac{1}{3}} + 20 \cos(x) - x^{-\frac{1}{20}} \][/tex]
3. Differentiate Each Term:
To find [tex]\(\frac{dy}{dx}\)[/tex], we differentiate each term separately.
- First Term: [tex]\( x^{-\frac{1}{3}} \)[/tex]
[tex]\[
\frac{d}{dx} \left( x^{-\frac{1}{3}} \right) = -\frac{1}{3} x^{-\frac{1}{3} - 1} = -\frac{1}{3} x^{-\frac{4}{3}}
\][/tex]
- Second Term: [tex]\( 20 \cos(x) \)[/tex]
[tex]\[
\frac{d}{dx} \left( 20 \cos(x) \right) = 20 \cdot (-\sin(x)) = -20 \sin(x)
\][/tex]
- Third Term: [tex]\( x^{-\frac{1}{20}} \)[/tex]
[tex]\[
\frac{d}{dx} \left( x^{-\frac{1}{20}} \right) = -\frac{1}{20} x^{-\frac{1}{20} - 1} = -\frac{1}{20} x^{-\frac{21}{20}}
\][/tex]
4. Combine the Results:
Now combine all the derivatives:
[tex]\[
\frac{dy}{dx} = -\frac{1}{3} x^{-\frac{4}{3}} - 20 \sin(x) - \frac{1}{20} x^{-\frac{21}{20}}
\][/tex]
Rewriting this gives:
[tex]\[
\frac{dy}{dx} = -\frac{1}{3} \cdot \frac{1}{x^{\frac{4}{3}}} - 20 \sin(x) - \frac{1}{20} \cdot \frac{1}{x^{\frac{21}{20}}}
\][/tex]
This is the derivative of the given function.
