6)Differentiate h (z) = 5z-3/z+ log, (2z) - 10 + sin(6z)
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![**Problem 6: Differentiation**
Differentiate the function \( h(x) = 5x^5 - 3\sqrt{x} + \log_3(2x) - 10^x + \sin(6x) \).
This problem requires the application of differentiation rules such as the power rule, chain rule, logarithmic differentiation, exponential differentiation, and the differentiation of trigonometric functions. Follow the procedures below to find the derivative \( h'(x) \):
1. **Power Rule** for \(5x^5\):
\[ \frac{d}{dx}[5x^5] = 5 \cdot 5x^{5-1} = 25x^4 \]
2. **Chain Rule** for \( -3\sqrt{x} \) where \(\sqrt{x} = x^{1/2}\):
\[ \frac{d}{dx}[-3x^{1/2}] = -3 \cdot \frac{1}{2}x^{(1/2)-1} = -\frac{3}{2}x^{-1/2} = -\frac{3}{2\sqrt{x}} \]
3. **Logarithmic Differentiation** for \( \log_3(2x) \):
\[ \frac{d}{dx}[\log_3(2x)] = \frac{1}{\ln(3)} \cdot \frac{d}{dx}[\ln(2x)] = \frac{1}{\ln(3)} \cdot \frac{1}{2x} \cdot 2 = \frac{1}{x\ln(3)} \]
4. **Exponential Differentiation** for \( -10^x \):
\[ \frac{d}{dx}[-10^x] = -10^x \cdot \ln(10) \]
5. **Differentiation of a Trigonometric Function** for \( \sin(6x) \):
\[ \frac{d}{dx}[\sin(6x)] = \cos(6x) \cdot 6 = 6\cos(6x) \]
Combining all the differentiated terms:
\[ h'(x) = 25x^4 - \frac{3}{2\sqrt{x}} + \frac](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb02fb262-35a8-4c2d-87d7-48cdf0e8c8b9%2F1d5241f6-b526-40ae-8528-f867c84fff47%2Fndqpj0l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 6: Differentiation**
Differentiate the function \( h(x) = 5x^5 - 3\sqrt{x} + \log_3(2x) - 10^x + \sin(6x) \).
This problem requires the application of differentiation rules such as the power rule, chain rule, logarithmic differentiation, exponential differentiation, and the differentiation of trigonometric functions. Follow the procedures below to find the derivative \( h'(x) \):
1. **Power Rule** for \(5x^5\):
\[ \frac{d}{dx}[5x^5] = 5 \cdot 5x^{5-1} = 25x^4 \]
2. **Chain Rule** for \( -3\sqrt{x} \) where \(\sqrt{x} = x^{1/2}\):
\[ \frac{d}{dx}[-3x^{1/2}] = -3 \cdot \frac{1}{2}x^{(1/2)-1} = -\frac{3}{2}x^{-1/2} = -\frac{3}{2\sqrt{x}} \]
3. **Logarithmic Differentiation** for \( \log_3(2x) \):
\[ \frac{d}{dx}[\log_3(2x)] = \frac{1}{\ln(3)} \cdot \frac{d}{dx}[\ln(2x)] = \frac{1}{\ln(3)} \cdot \frac{1}{2x} \cdot 2 = \frac{1}{x\ln(3)} \]
4. **Exponential Differentiation** for \( -10^x \):
\[ \frac{d}{dx}[-10^x] = -10^x \cdot \ln(10) \]
5. **Differentiation of a Trigonometric Function** for \( \sin(6x) \):
\[ \frac{d}{dx}[\sin(6x)] = \cos(6x) \cdot 6 = 6\cos(6x) \]
Combining all the differentiated terms:
\[ h'(x) = 25x^4 - \frac{3}{2\sqrt{x}} + \frac
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