14. [-/1.09 Points] DETAILS MY NOTES SCALCET9 $$ 3 \times \mathrm{XPA} $$ Use logarithmic differentiation to find the derivative of the function. $$ y=x^{4} \cos (x) $$ $$ y^{\prime}=\square $$

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Let $$ y = x^4 \cos(x) $$.

1. Take the natural logarithm of both sides:
   $$
   \ln(y) = \ln\left(x^4 \cos(x)\right)
   $$
   
2. Use the logarithm property $$\ln(ab) = \ln(a) + \ln(b)$$:
   $$
   \ln(y) = \ln(x^4) + \ln\left(\cos(x)\right)
   $$
   
3. Simplify using the power rule $$\ln(x^4) = 4\ln(x)$$:
   $$
   \ln(y) = 4\ln(x) + \ln\left(\cos(x)\right)
   $$
   
4. Differentiate both sides with respect to $$ x $$. For the left-hand side, use the chain rule:
   $$
   \frac{1}{y} y' = \frac{4}{x} + \left(\frac{1}{\cos(x)}\right)(-\sin(x))
   $$
   
5. Simplify the derivative on the right-hand side:
   $$
   \frac{1}{y} y' = \frac{4}{x} - \tan(x)
   $$
   
6. Solve for $$ y' $$ by multiplying both sides by $$ y $$:
   $$
   y' = y\left(\frac{4}{x} - \tan(x)\right)
   $$
   
7. Substitute back $$ y = x^4 \cos(x) $$:
   $$
   y' = x^4 \cos(x) \left(\frac{4}{x} - \tan(x)\right)
   $$
   
8. Simplify further:
   $$
   y' = 4x^3 \cos(x) - x^4 \cos(x)\tan(x)
   $$
   
9. Note that $$ \cos(x) \tan(x) = \sin(x) $$:
   $$
   y' = 4x^3 \cos(x) - x^4 \sin(x)
   $$

$$
y' = 4x^3 \cos(x) - x^4 \sin(x)
$$
$$ y' = 4x^3 \cos(x) - x^4 \sin(x) $$

[-/1.09 Points] DETAILS MY NOTES SCALCET9
Use logarithmic differentiation to find the derivative of the function.

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[-/1.09 Points] DETAILS MY NOTES SCALCET9 Use logarithmic differentiation to find the derivative of the function.