If y equals open parentheses cos space x close parentheses to the power of x squared end exponent comma spacethen fraction numerator d y over denominator d x end fraction equals

a.
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b.
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c.
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d.
negative x space open parentheses cos space x close parentheses to the power of x squared end exponent open parentheses x space tan space x space minus 2 space ln left parenthesis cos space x right parenthesis close parentheses

e.
negative x space open parentheses cos space x close parentheses to the power of x squared end exponent open parentheses x space tan space x space plus 2 space ln left parenthesis cos space x right parenthesis close parentheses

To find \(\frac{dy}{dx}\) where \(y = (\cos x)^{x^2}\), we will use logarithmic differentiation.

Given:
\[ y = (\cos x)^{x^2} \]

1. Take the natural logarithm of both sides:
\[ \ln y = \ln \left( (\cos x)^{x^2} \right) \]
\[ \ln y = x^2 \ln (\cos x) \]

2. Differentiate both sides with respect to \(x\):
\[ \frac{d}{dx} (\ln y) = \frac{d}{dx} (x^2 \ln (\cos x)) \]

Using the chain rule on the left side and the product rule on the right side:
\[ \frac{1}{y} \frac{dy}{dx} = 2x \ln (\cos x) + x^2 \cdot \frac{d}{dx} (\ln (\cos x)) \]

3. The derivative of \(\ln (\cos x)\) is:
\[ \frac{d}{dx} (\ln (\cos x)) = \frac{1}{\cos x} \cdot (-\sin x) = -\tan x \]

Thus,
\[ \frac{1}{y} \frac{dy}{dx} = 2x \ln (\cos x) + x^2 (-\tan x) \]
\[ \frac{1}{y} \frac{dy}{dx} = 2x \ln (\cos x) - x^2 \tan x \]

4. Multiply both sides by \(y\) to solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = y \left( 2x \ln (\cos x) - x^2 \tan x \right) \]

Substitute \(y = (\cos x)^{x^2}\):
\[ \frac{dy}{dx} = (\cos x)^{x^2} \left( 2x \ln (\cos x) - x^2 \tan x \right) \]

Comparing this with the given options, we have:

a. \((\cos x)^{x^2} (-2x \tan x + x^2 \ln (\cos x))\)

b. \(x^2 (\cos x)^{x^2-1} (-\sin x) (2x)\)

c. \(-x^2 (\cos x)^{x^2} (\tan x - \ln (\cos x))\)

d. \(-x (\cos x)^{x^2} (x \tan x - 2 \ln (\cos x))\)

e. \(-x (\cos x)^{x^2} (x \tan x + 2 \ln (\cos x))\)

The correct answer matches option (a).

\[ \boxed{a} \]

Would you like more details or have any questions? Here are 5 more questions you can ask:
1. Can you explain the product rule in calculus?
2. What is the chain rule and how is it used in differentiation?
3. How do you find the derivative of a function raised to a power?
4. What is logarithmic differentiation and when is it used?
5. Can you solve another example involving logarithmic differentiation?

**Tip:** When solving complex differentiation problems, breaking them down using rules like the chain rule, product rule, and logarithmic differentiation can simplify the process and reduce errors.

Learn how to solve the logarithmic differentiation problem for y = (cos x)^(x^2). This problem involves advanced calculus concepts such as logarithmic differentiation, the chain rule, and the product rule. Detailed step-by-step solution provided.

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