To differentiate $f(x) = x^{2\cos(x)}$, we'll use logarithmic differentiation because the variable $x$ is both the base and appears in the exponent. Here's the step-by-step process:

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Step 1: Take the natural logarithm

Take the natural logarithm of both sides: $\ln(f(x)) = \ln\left(x^{2\cos(x)}\right)$

Using the logarithmic property $\ln(a^b) = b \ln(a)$, this becomes: $\ln(f(x)) = 2\cos(x) \ln(x)$

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Step 2: Differentiate implicitly

Differentiate both sides with respect to $x$. Remember to apply the product rule to $2\cos(x) \ln(x)$ on the right-hand side: $\frac{f'(x)}{f(x)} = \frac{d}{dx} \left( 2\cos(x) \ln(x) \right)$

The derivative of $2\cos(x) \ln(x)$ is: $\frac{d}{dx} \left( 2\cos(x) \ln(x) \right) = 2(-\sin(x))\ln(x) + 2\cos(x)\frac{1}{x}$

So: $\frac{f'(x)}{f(x)} = -2\sin(x)\ln(x) + \frac{2\cos(x)}{x}$

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Step 3: Solve for $f'(x)$

Multiply through by $f(x) = x^{2\cos(x)}$: $f'(x) = x^{2\cos(x)} \left( -2\sin(x)\ln(x) + \frac{2\cos(x)}{x} \right)$

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Final Answer:

The derivative of $f(x) = x^{2\cos(x)}$ is: $f'(x) = x^{2\cos(x)} \left( -2\sin(x)\ln(x) + \frac{2\cos(x)}{x} \right)$

Would you like to see this expressed in more compact notation, or do you have further questions?

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5 Related Questions:

1. How do we apply logarithmic differentiation to other functions involving both variable bases and exponents?
2. What are common scenarios where logarithmic differentiation is necessary?
3. How can we use the product rule effectively in compound differentiation problems?
4. Can you provide an example of integrating functions with variable bases and exponents?
5. How does $x^{a(x)}$ compare to $a(x)^{x}$ in terms of differentiation?

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Tip:

When faced with functions involving variables in both the base and exponent, logarithmic differentiation often simplifies the process significantly.