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.

Question

UpStudy AI · Accepted Answer

**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.
$$ f^{\prime}(x) = 5\cos(x) + \frac{1}{x} $$ 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.

If , find

Graph and in the same viewing window.

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