Find the derivative of f left parenthesis x right parenthesis equals 2 over x to the power of 6 minus 4 cos x minus fifth root of x.


f apostrophe left parenthesis x right parenthesis equals short dash 12 over x to the power of 7 minus 4 sin x minus fraction numerator 1 over denominator 5 x to the power of begin display style 4 over 5 end style end exponent end fraction


f apostrophe left parenthesis x right parenthesis equals fraction numerator 1 over denominator 3 x to the power of 5 end fraction minus 4 sin x minus fraction numerator 1 over denominator 5 x to the power of begin display style 4 over 5 end style end exponent end fraction


f apostrophe left parenthesis x right parenthesis equals fraction numerator 1 over denominator 3 x to the power of 5 end fraction plus 4 sin x minus fraction numerator 1 over denominator 5 x to the power of begin display style 4 over 5 end style end exponent end fraction


f apostrophe left parenthesis x right parenthesis equals short dash 12 over x to the power of 7 plus 4 sin x minus fraction numerator 1 over denominator 5 x to the power of begin display style 4 over 5 end style end exponent end fraction

To find the derivative of the function \( f(x) = \frac{2}{x^6} - 4\cos(x) - \sqrt[5]{x} \), we will use the rules of differentiation. Specifically, we will use the power rule, the chain rule, and the derivative of the cosine function.

Given function:
\[ f(x) = \frac{2}{x^6} - 4\cos(x) - \sqrt[5]{x} \]

Rewrite \(\frac{2}{x^6}\) as \(2x^{-6}\) and \(\sqrt[5]{x}\) as \(x^{1/5}\).

\[ f(x) = 2x^{-6} - 4\cos(x) - x^{1/5} \]

Now, differentiate term by term:

1. For \(2x^{-6}\):
\[ \frac{d}{dx}[2x^{-6}] = 2 \cdot (-6)x^{-7} = -12x^{-7} \]

2. For \(-4\cos(x)\):
\[ \frac{d}{dx}[-4\cos(x)] = -4 \cdot (-\sin(x)) = 4\sin(x) \]

3. For \(-x^{1/5}\):
\[ \frac{d}{dx}[-x^{1/5}] = -\frac{1}{5}x^{1/5-1} = -\frac{1}{5}x^{-4/5} \]

Combining these results, we get:
\[ f'(x) = -12x^{-7} + 4\sin(x) - \frac{1}{5}x^{-4/5} \]

Rewrite this in a more conventional form:
\[ f'(x) = -\frac{12}{x^7} + 4\sin(x) - \frac{1}{5}x^{-4/5} \]

Comparing with the given choices, the correct one is:

\[ f'(x) = -\frac{12}{x^7} + 4\sin(x) - \frac{1}{5}x^{-4/5} \]

Thus, the correct option is:
\[ f'(x) = -\frac{12}{x^7} + 4\sin(x) - \frac{1}{5}x^{-4/5} \]

Do you want more details or have any questions?

Here are 5 more questions you might want to ask:
1. How do you integrate the function \(f(x) = \frac{2}{x^6} - 4\cos(x) - \sqrt[5]{x}\)?
2. What are the critical points of the function \(f(x) = \frac{2}{x^6} - 4\cos(x) - \sqrt[5]{x}\)?
3. How do you find the second derivative of \(f(x) = \frac{2}{x^6} - 4\cos(x) - \sqrt[5]{x}\)?
4. Can you explain the chain rule in differentiation with examples?
5. How do you determine the concavity of the function \(f(x) = \frac{2}{x^6} - 4\cos(x) - \sqrt[5]{x}\)?

Tip: When differentiating functions, always break them down into simpler components and apply differentiation rules systematically.

Learn how to find the derivative of the function f(x) = 2/x^6 - 4cos(x) - 5th root of x step-by-step with detailed explanations. This problem covers differentiation rules including the power rule and chain rule, and involves trigonometric and root functions. Suitable for advanced high school students and calculus enthusiasts.

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