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Find [tex]\(\frac{dy}{dx}\)[/tex] for the following function:

[tex]\[
a = y = \frac{1}{\sqrt[3]{x}} + 20 \cos \left(-\frac{\sqrt[5]{x}}{\sqrt[4]{x}}\right)
\][/tex]

Sagot :

GinnyAnsweridn
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.
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