If 𝑥𝑥, determine the 𝑑𝑦𝑑𝑥Question 8Answera.1+𝑙𝑛𝑥b.𝑥𝑥𝑙𝑛𝑥c.𝑥(1+𝑥𝑙𝑛𝑥)d.𝑥𝑥(1+𝑙𝑛𝑥)

It seems like there's a mistake in your question as it's not clear what function you want to differentiate. However, I can show you how to differentiate the functions you've mentioned in the options.

a. The derivative of 1+ln(x) with respect to x is 1/x. This is because the derivative of ln(x) is 1/x and the derivative of a constant is 0.

b. The derivative of xln(x) with respect to x is 1ln(x) + x*(1/x) = ln(x) + 1. This is obtained using the product rule.

c. The derivative of x*(1+xln(x)) with respect to x is 1(1+xln(x)) + x(1+ln(x)) = 1 + x*ln(x) + 1 + ln(x). This is obtained using the product rule.

d. The derivative of xx(1+ln(x)) with respect to x is 2x(1+ln(x)) + xx(1/x) = 2x + 2x*ln(x) + x. This is obtained using the product rule.