To find the derivative of the function \( y = (\cos x)^x + x^{\log x} \), we need to apply the rules of differentiation, including the chain rule, product rule, and logarithmic differentiation.

### 1. Differentiate \( y = (\cos x)^x \):

Let \( u(x) = (\cos x)^x \). To differentiate this, use logarithmic differentiation.

\[
\ln u(x) = \ln \left( (\cos x)^x \right)
\]
\[
\ln u(x) = x \ln(\cos x)
\]

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

\[
\frac{1}{u(x)} \frac{du(x)}{dx} = \ln(\cos x) + x \cdot \frac{-\sin x}{\cos x}
\]
\[
\frac{du(x)}{dx} = u(x) \left( \ln(\cos x) - x \tan x \right)
\]
\[
\frac{du(x)}{dx} = (\cos x)^x \left( \ln(\cos x) - x \tan x \right)
\]

### 2. Differentiate \( y = x^{\log x} \):

Let \( v(x) = x^{\log x} \). Again, use logarithmic differentiation.

\[
\ln v(x) = \log x \cdot \ln x
\]

Differentiate both sides with respect to \( x \):

\[
\frac{1}{v(x)} \frac{dv(x)}{dx} = \frac{1}{x} \ln x + \frac{\log x}{x}
\]
\[
\frac{dv(x)}{dx} = x^{\log x} \left( \frac{\ln x + \log x}{x} \right)
\]

### 3. Combine the derivatives:

Now, the derivative of the entire function \( y \) is:

\[
\frac{dy}{dx} = \frac{du(x)}{dx} + \frac{dv(x)}{dx}
\]
\[
\frac{dy}{dx} = (\cos x)^x \left( \ln(\cos x) - x \tan x \right) + x^{\log x} \left( \frac{\ln x + \log x}{x} \right)
\]

This is the derivative of the function \( y = (\cos x)^x + x^{\log x} \).

Would you like a step-by-step breakdown or further explanation on any of the differentiation techniques used here?

Here are some related questions:

1. How would you differentiate \( y = (\sin x)^x \)?
2. What is the derivative of \( y = x^{\sin x} \)?
3. Can you find the second derivative of \( y = (\cos x)^x + x^{\log x} \)?
4. How do you apply logarithmic differentiation to a function of the form \( y = a^x \)?
5. What is the derivative of \( y = e^{x \ln x} \)?

**Tip:** Logarithmic differentiation is a powerful technique when dealing with functions where the variable is both the base and the exponent.

Learn how to find the derivative of y = (cos x)^x + x^(log x) using logarithmic differentiation and other differentiation rules. This problem involves advanced calculus techniques suitable for high school students studying differentiation.

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