Question

Solve dy/dx of y=(x-1)diveide(x+1)
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Answer 1

I assume you're reffering to 5.b.

If

`y = \left((x-1)/(x+1)\right)^(1/3)`

then we can rewrite this as

`y^3 = (x-1)/(x+1) \\\\ \ln(y^3) = \ln\left((x-1)/(x+1)\right) \\\\ 3\ln(y) = \ln(x-1) - \ln(x+1)`

Now differentiate both sides implicitly:

`(\mathrm d)/(\mathrm dx)\left[3\ln(y(x))\right] = (\mathrm d)/(\mathrm dx)\left[\ln(x-1)-\ln(x+1)\right] \\\\ \frac3{y(x)}(\mathrm dy)/(\mathrm dx) = \frac1{x-1} - \frac1{x+1} \\\\ (\mathrm dy)/(\mathrm dx) = \frac{y(x)}3\left(\frac1{x-1}-\frac1{x+1}\right) \\\\ (\mathrm dy)/(\mathrm dx) = \frac13 \left((x-1)/(x+1)\right)^(1/3) \left(\frac1{x-1}-\frac1{x+1}\right) \\\\ (\mathrm dy)/(\mathrm dx) = \boxed{\frac23 \frac1{x^2-1}\left((x-1)/(x+1)\right)^(1/3)}`