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If \( f(x)=5 \sin (x)+\ln (3 x) \), find \( f^{\prime}(x) \) \( f^{\prime}(x)= \) Graph \( f \) and \( f^{\prime} \) in the same viewing window.

Ask by Chang Daniels. in the United States
Mar 20,2025

Solución de inteligencia artificial de Upstudy

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\( f^{\prime}(x) = 5\cos(x) + \frac{1}{x} \) To graph \( f(x) = 5\sin(x) + \ln(3x) \) and \( f^{\prime}(x) = 5\cos(x) + \frac{1}{x} \) in the same viewing window, use the following Python code: ```python import numpy as np import matplotlib.pyplot as plt def f(x): return 5 * np.sin(x) + np.log(3 * x) def fp(x): return 5 * np.cos(x) + 1 / x x = np.linspace(0.1, 10, 400) y1 = f(x) y2 = fp(x) plt.figure(figsize=(10, 6)) plt.plot(x, y1, label=r'$f(x)=5\sin(x)+\ln(3x)$', color='blue') plt.plot(x, y2, label=r"$f'(x)=5\cos(x)+\frac{1}{x}$", color='red', linestyle='--') plt.title('Graph of $f(x)$ and $f^{\prime}(x)$') plt.xlabel('x') plt.ylabel('y') plt.legend() plt.grid(True) plt.show() ``` This code will display both functions on the same graph.

Solución

**Step 1. Differentiate \( f(x)=5\sin(x)+\ln(3x) \).** Differentiate each term: 1. The derivative of \( 5\sin(x) \) is: \[ 5\cos(x) \] 2. The derivative of \( \ln(3x) \) uses the chain rule. First, recall that: \[ \frac{d}{dx}\ln(u) = \frac{1}{u}\frac{du}{dx} \] For \( u=3x \), we have \( \frac{du}{dx}=3 \). Therefore: \[ \frac{d}{dx}\ln(3x)=\frac{1}{3x}\cdot3=\frac{1}{x} \] Thus, the derivative \( f^{\prime}(x) \) is: \[ f^{\prime}(x)=5\cos(x)+\frac{1}{x} \] --- **Step 2. Graph \( f(x) \) and \( f^{\prime}(x) \) in the same viewing window.** Below is a sample Python code using Matplotlib to graph both functions. Note that the domain must be \( x>0 \) since \( \ln(3x) \) is defined only for positive \( x \). ```python import numpy as np import matplotlib.pyplot as plt # Define the functions def f(x): return 5 * np.sin(x) + np.log(3 * x) def fp(x): return 5 * np.cos(x) + 1 / x # Create an array of x values, avoiding x = 0 x = np.linspace(0.1, 10, 400) # Compute y values for f(x) and f'(x) y1 = f(x) y2 = fp(x) # Create the plot plt.figure(figsize=(10, 6)) plt.plot(x, y1, label=r'$f(x)=5\sin(x)+\ln(3x)$', color='blue') plt.plot(x, y2, label=r"$f'(x)=5\cos(x)+\frac{1}{x}$", color='red', linestyle='--') plt.title('Graph of $f(x)$ and $f^{\prime}(x)$') plt.xlabel('x') plt.ylabel('y') plt.legend() plt.grid(True) plt.show() ``` When you run this code in a Python environment with Matplotlib installed, it will produce a graph showing both \( f(x) \) and \( f^{\prime}(x) \) on the same viewing window.

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Mind Expander

To find \( f^{\prime}(x) \), you'll use the rules of differentiation. The derivative of \( f(x) = 5 \sin (x) + \ln (3x) \) can be computed as follows: 1. The derivative of \( 5 \sin(x) \) is \( 5 \cos(x) \) using the chain and basic derivative rules. 2. The derivative of \( \ln(3x) \) is \( \frac{3}{3x} = \frac{1}{x} \) using the derivative of the natural logarithm and the constant multiple rule. Putting it all together, you get: \[ f^{\prime}(x) = 5 \cos(x) + \frac{1}{x} \] Now, for graphing \( f \) and \( f^{\prime} \): Graph \( f(x) = 5 \sin(x) + \ln(3x) \) and its derivative \( f^{\prime}(x) = 5 \cos(x) + \frac{1}{x} \) in the same viewing window, for example, you could set the x-axis from \( 0.1 \) to \( 10 \) and the y-axis from \( -5 \) to \( 10 \). This will allow you to appreciate the oscillations of the sine and cosine functions together with the behavior of the logarithmic term as \( x \) increases!

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