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.
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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!
