Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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do not simplify the answer
![The given function is:
\( f(x) = 2x^7 + \frac{7}{x^2} + \frac{2}{7}x - 2^7 + \sqrt[7]{x^2} \)
To find the derivative \( f'(x) \), use the following rules for differentiation:
1. **Power Rule**: For any term of the form \( ax^n \), the derivative is \( anx^{n-1} \).
2. **Reciprocal Rule**: For a term of the form \( \frac{a}{x^n} \), rewrite it as \( ax^{-n} \) and then apply the power rule.
3. **Constant**: The derivative of a constant term is zero.
4. **Fractional Exponents**: For terms like \(\sqrt[n]{x^m}\), rewrite as \(x^{m/n}\) and use the power rule.
The derivative of the function is:
1. \( \frac{d}{dx}[2x^7] = 14x^6 \)
2. \( \frac{d}{dx}[\frac{7}{x^2}] = \frac{d}{dx}[7x^{-2}] = -14x^{-3} \)
3. \( \frac{d}{dx}[\frac{2}{7}x] = \frac{2}{7} \)
4. The derivative of the constant term \(-2^7\) is 0.
5. \( \frac{d}{dx}[\sqrt[7]{x^2}] = \frac{d}{dx}[x^{2/7}] = \frac{2}{7}x^{-5/7} \)
Putting it all together:
\( f'(x) = 14x^6 - 14x^{-3} + \frac{2}{7} + \frac{2}{7}x^{-5/7} \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fb7327e-adf7-4bbc-a708-e987be622d92%2Fb44f8554-e3ca-4d2a-b4a1-663ac69cd625%2Fuko2wbd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The given function is:
\( f(x) = 2x^7 + \frac{7}{x^2} + \frac{2}{7}x - 2^7 + \sqrt[7]{x^2} \)
To find the derivative \( f'(x) \), use the following rules for differentiation:
1. **Power Rule**: For any term of the form \( ax^n \), the derivative is \( anx^{n-1} \).
2. **Reciprocal Rule**: For a term of the form \( \frac{a}{x^n} \), rewrite it as \( ax^{-n} \) and then apply the power rule.
3. **Constant**: The derivative of a constant term is zero.
4. **Fractional Exponents**: For terms like \(\sqrt[n]{x^m}\), rewrite as \(x^{m/n}\) and use the power rule.
The derivative of the function is:
1. \( \frac{d}{dx}[2x^7] = 14x^6 \)
2. \( \frac{d}{dx}[\frac{7}{x^2}] = \frac{d}{dx}[7x^{-2}] = -14x^{-3} \)
3. \( \frac{d}{dx}[\frac{2}{7}x] = \frac{2}{7} \)
4. The derivative of the constant term \(-2^7\) is 0.
5. \( \frac{d}{dx}[\sqrt[7]{x^2}] = \frac{d}{dx}[x^{2/7}] = \frac{2}{7}x^{-5/7} \)
Putting it all together:
\( f'(x) = 14x^6 - 14x^{-3} + \frac{2}{7} + \frac{2}{7}x^{-5/7} \)
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