**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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Answered: 6)Differentiate h (z) = 5z-3/z+ log,…

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Trigonometry

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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Concept explainers

Maximization

Have you ever seen a bridge in the shape of a parabola? Where is its maximum on the bridge? The maximum of that bridge is the highest point on that bridge. The process of finding the maximum of the bridge here is called the maximization.

Question

**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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