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To find the derivative of the given function \( y = x^{4\cos(x)} \) using logarithmic differentiation, we proceed as follows:

1. Take the natural logarithm of both sides:
\[
\ln y = \ln(x^{4\cos(x)}).
\]

2. Simplify the right-hand side using the logarithm property \( \ln(a^b) = b\ln a \):
\[
\ln y = 4\cos(x) \ln x.
\]

3. Differentiate both sides with respect to \( x \):
\[
\frac{d}{dx}(\ln y) = \frac{d}{dx} [4\cos(x) \ln x].
\]

The left-hand side becomes:
\[
\frac{1}{y} \frac{dy}{dx}.
\]

The right-hand side requires using the product rule:
\[
\frac{d}{dx} [4\cos(x) \ln x] = 4 \frac{d}{dx}(\cos(x)) \ln x + 4\cos(x) \frac{d}{dx}(\ln x).
\]

Calculating each derivative:
- \(\frac{d}{dx}(\cos(x)) = -\sin(x)\),
- \(\frac{d}{dx}(\ln x) = \frac{1}{x}\).

So, the right-hand side becomes:
\[
4(-\sin(x)) \ln x + 4\cos(x) \cdot \frac{1}{x}.
\]

Putting it all together:
\[
\frac{1}{y} \frac{dy}{dx} = 4(-\sin(x)) \ln x + \frac{4\cos(x)}{x}.
\]

Multiply both sides by \( y \) to solve for \( \frac{dy}{dx} \):
\[
\frac{dy}{dx} = y \left[4(-\sin(x)) \ln x + \frac{4\cos(x)}{x}\right].
\]

Since \( y = x^{4\cos(x)} \), substitute it back in:
\[
\frac{dy}{dx} = x^{4\cos(x)} \left[4(-\sin(x)) \ln x + \frac{4\cos(x)}{x}\right].
\]

The derivative of the function is:
\[
y' = x^{4\cos(x)} \left[\frac{4\cos(x)}{x} - 4\sin(x) \ln x\right].
\]

Would you like any further details or have any questions about this solution?

Here are five related questions to deepen your understanding:

1. How do you apply logarithmic differentiation to functions involving products and exponents?
2. What other methods could be used to find the derivative of exponential functions?
3. How does the product rule apply in more complex differentiation problems?
4. Why is it beneficial to use logarithmic differentiation for functions like \( x^{4\cos(x)} \)?
5. How can logarithmic differentiation simplify the process of finding derivatives in calculus?

**Tip:** When using logarithmic differentiation, taking the natural log of both sides first helps break down complicated expressions, making differentiation more manageable.

Use logarithmic differentiation to find the derivative of the function y = x^(4cos(x)).

Explore how to find the derivative of the function y = x^(4cos(x)) using logarithmic differentiation. This step-by-step guide breaks down complex differentiation using calculus rules and properties.

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