Final answer to the problem
$\left(\ln\left(x\right)+1\right)x^x\ln\left(x\right)^{\cos\left(2x\right)}+x^x\left(-2\sin\left(2x\right)\ln\left(\ln\left(x\right)\right)+\frac{\cos\left(2x\right)}{x\ln\left(x\right)}\right)\ln\left(x\right)^{\cos\left(2x\right)}$
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Step-by-step Solution
Learn how to solve product rule of differentiation problems step by step online. Find the derivative of x^xln(x)^cos(2x). Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=x^x and g=\ln\left(x\right)^{\cos\left(2x\right)}. The derivative \frac{d}{dx}\left(x^x\right) results in \left(\ln\left(x\right)+1\right)x^x. The derivative \frac{d}{dx}\left(\ln\left(x\right)^{\cos\left(2x\right)}\right) results in \left(-2\sin\left(2x\right)\ln\left(\ln\left(x\right)\right)+\frac{\cos\left(2x\right)}{x\ln\left(x\right)}\right)\ln\left(x\right)^{\cos\left(2x\right)}.
Final answer to the problem
$\left(\ln\left(x\right)+1\right)x^x\ln\left(x\right)^{\cos\left(2x\right)}+x^x\left(-2\sin\left(2x\right)\ln\left(\ln\left(x\right)\right)+\frac{\cos\left(2x\right)}{x\ln\left(x\right)}\right)\ln\left(x\right)^{\cos\left(2x\right)}$
