Given the function $f\left(x\right)=\sin \left(x\right)\ln \left(x^4\right)$, compute $f^{\prime }\left(x\right)$. Which of the following is correct?

• A. $f^{\prime }\left(x\right)=\cos \left(x\right)\ln \left(x\right)+\frac{4}{x}\sin \left(x\right)$
• B. $f^{\prime }\left(x\right)=\cos \left(x\right)\ln \left(x^4\right)+\frac{4\sin \left(x\right)}{x}$
• C. $f^{\prime }\left(x\right)=\sin \left(x\right)\frac{4}{x}-\cos \left(x\right)\ln \left(x^4\right)$
• D. $f^{\prime }\left(x\right)=\cos \left(x\right)\ln \left(x^4\right)-\frac{4\sin \left(x\right)}{x}$

Answer: (B)

Verified step by step guidance

1. Step 1: Recognize that the function f(x) = sin(x) * ln(x^4) is a product of two functions: u(x) = sin(x) and v(x) = ln(x^4). To compute f'(x), we will use the product rule, which states that if f(x) = u(x) * v(x), then f'(x) = u'(x) * v(x) + u(x) * v'(x).
2. Step 2: Compute u'(x), the derivative of u(x) = sin(x). The derivative of sin(x) is cos(x), so u'(x) = cos(x).
3. Step 3: Compute v'(x), the derivative of v(x) = ln(x^4). First, simplify ln(x^4) using the logarithmic property ln(a^b) = b * ln(a). This gives v(x) = 4 * ln(x). Now, differentiate v(x) = 4 * ln(x). The derivative of ln(x) is 1/x, so v'(x) = 4/x.
4. Step 4: Apply the product rule. Substitute u(x), u'(x), v(x), and v'(x) into the formula f'(x) = u'(x) * v(x) + u(x) * v'(x). This gives f'(x) = cos(x) * ln(x^4) + sin(x) * (4/x).
5. Step 5: Simplify the expression. The final derivative is f'(x) = cos(x) * ln(x^4) + (4 * sin(x))/x. Compare this result with the given options to identify the correct answer.