Final answer to the problem
$\left(\ln\left(x\right)+1\right)x^xy+x^xy^{\prime}=2\left(-\sin\left(x\right)\ln\left(x\right)+\frac{\cos\left(x\right)}{x}\right)x^{2\cos\left(x\right)}$
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Step-by-step Solution
Learn how to solve product rule of differentiation problems step by step online. Find the implicit derivative d/dx(x^xy=x^(2cos(x))). Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable. Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=x^x and g=y. The derivative of the linear function is equal to 1. The derivative \frac{d}{dx}\left(x^x\right) results in \left(\ln\left(x\right)+1\right)x^x.
Final answer to the problem
$\left(\ln\left(x\right)+1\right)x^xy+x^xy^{\prime}=2\left(-\sin\left(x\right)\ln\left(x\right)+\frac{\cos\left(x\right)}{x}\right)x^{2\cos\left(x\right)}$
