<s>[INST] Find derivative of tricky logarithmic functions Find the derivative of $y=(x^{x+1})(x+1)^x$
So this is what I have,
$$\ln y=\ln[(x^{x+1})(x+1)^x]$$
$$= \ln x^{x+1} + \ln(x+1)^x$$
$$\frac{1}{y}y' = (1)(\ln x) + (x+1)\frac{1}{x} + (1)(\ln(x+1)) + (x)\frac{1}{x+1}$$
$$= \ln x + \frac{x+1}{x} + \ln(x+1) + \frac{x}{x+1}$$
$$y' = (x^{x+1})(x+1)^x \left[\ln x + \ln(x+1) + \frac{x+1}{x} + \frac{x}{x+1}\right]$$
Is that the right answer?
 [/INST] Yes that is the right answer..
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