Final answer to the problem
$\left(\ln\left(x\right)+1\right)x^x\left(1+7x\right)^{\frac{3}{x}}+3\left(7x-\ln\left(1+7x\right)-7x\ln\left(1+7x\right)\right)\left(1+7x\right)^{\left(\frac{3}{x}-1\right)}x^{\left(x-2\right)}$
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Step-by-step Solution
Learn how to solve problems step by step online. Find the derivative of x^x(1+7x)^(3/x). Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=x^x and g=\left(1+7x\right)^{\frac{3}{x}}. 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(\left(1+7x\right)^{\frac{3}{x}}\right) results in \frac{3\left(7x-\ln\left(1+7x\right)-7x\ln\left(1+7x\right)\right)\left(1+7x\right)^{\left(\frac{3}{x}-1\right)}}{x^2}. Simplify the derivative.
Final answer to the problem
$\left(\ln\left(x\right)+1\right)x^x\left(1+7x\right)^{\frac{3}{x}}+3\left(7x-\ln\left(1+7x\right)-7x\ln\left(1+7x\right)\right)\left(1+7x\right)^{\left(\frac{3}{x}-1\right)}x^{\left(x-2\right)}$
