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

b.
x squared space open parentheses cos space x close parentheses to the power of x squared minus 1 end exponent open parentheses negative sin space x close parentheses left parenthesis 2 x right parenthesis

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

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

Given the function \( y = (\cos x)^{x^2} \), we need to find its derivative \(\frac{dy}{dx}\).

To solve this, we will use logarithmic differentiation. Let's start by taking the natural logarithm of both sides:

\[
\ln y = \ln \left( (\cos x)^{x^2} \right)
\]

Using the logarithm power rule \(\ln(a^b) = b \ln a\), we get:

\[
\ln y = x^2 \ln (\cos x)
\]

Next, differentiate both sides with respect to \(x\):

\[
\frac{d}{dx} (\ln y) = \frac{d}{dx} \left( x^2 \ln (\cos x) \right)
\]

On the left side, using the chain rule:

\[
\frac{d}{dx} (\ln y) = \frac{1}{y} \frac{dy}{dx}
\]

On the right side, using the product rule \(\frac{d}{dx} (u \cdot v) = u'v + uv'\), where \(u = x^2\) and \(v = \ln (\cos x)\):

\[
\frac{d}{dx} \left( x^2 \ln (\cos x) \right) = 2x \ln (\cos x) + x^2 \cdot \frac{d}{dx} (\ln (\cos x))
\]

To differentiate \(\ln (\cos x)\):

\[
\frac{d}{dx} (\ln (\cos x)) = \frac{1}{\cos x} \cdot (-\sin x) = -\tan x
\]

Therefore:

\[
\frac{d}{dx} \left( x^2 \ln (\cos x) \right) = 2x \ln (\cos x) + x^2 (-\tan x)
\]

Combining everything:

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

Multiplying both sides by \(y\):

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

Recall that \(y = (\cos x)^{x^2}\):

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

Rewriting the expression:

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

Comparing with the given options, the correct answer is:

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

Learn how to find the derivative of (cos x)^(x^2) using logarithmic differentiation with a step-by-step solution. This problem covers advanced differentiation techniques, including the logarithm power rule, chain rule, and product rule. Suitable for advanced high school students or college-level calculus students.

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