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Use logarithmic differentiation to find the derivative of the function y=x²x

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Final answer:

To find the derivative of the function y = x²x using logarithmic differentiation, we can take the natural logarithm of both sides of the equation and apply the properties of logarithms. Differentiating both sides using the chain rule, we can derive the equation for the derivative of y. The derivative of y = x²x is y' = (2/x) + 2x/y.

Step-by-step explanation:

To find the derivative of the function y = x²x using logarithmic differentiation, we can take the natural logarithm of both sides of the equation. So, ln(y) = ln(x²x). Next, we can use the properties of logarithms to simplify the equation. Applying the power rule, we get ln(y) = ln(x²) + ln(x). Then, we can differentiate both sides of the equation with respect to x using the chain rule. The derivative of ln(y) with respect to x is (1/y) * y'. The derivative of ln(x²) with respect to x is (1/x²) * (2x). The derivative of ln(x) with respect to x is 1/x.

Combining these results, we have (1/y) * y' = (1/x²) * (2x) + (1/x). Multiplying through by y and simplifying, we obtain y' = (2/x) + 2x/y. Therefore, the derivative of y = x²x is y' = (2/x) + 2x/y.

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