18.01 OCW question 1F-3. Implicit differentiation

Question

jkristia · Answer

Got a question regarding 1F-3 where we are asked to find $dy/dx$ using implicit differentiation.
$$y = x^n$$
First I don't understand why not just use regular differentiation here as it gives you the correct result and is simpler than trying implicit, but anyway, the problem is regarding implicit.\

I'm fine until the last step, so I have 

$$y' = \frac{1}{n} y^{(1-n)}$$

And then it get replaced with 

$$y' = \frac{1}{n} x^{(\frac{1}{n}-1)}$$

Which is the same result as I get with explicit differentiation, but I fail to see how this substitution is done. I know $y = x^n$, but still, something is missing

jkristia · Answer

Ahh, got it
$$y = x^{\frac{1}{n}}$$
$$y^n = x$$
$$y' = \frac{1}{n} y^{(1-n)}$$
$$y' = \frac{1}{n} \frac{y^1}{y^{n}} $$
$$y' = \frac{1}{n} \frac{x^{\frac{1}{n}}}{y^{n}} $$
$$y' = \frac{1}{n} \frac{x^{\frac{1}{n}}}{(x^{\frac{1}{n}})^{n}} $$
$$y' = \frac{1}{n} \frac{x^{\frac{1}{n}}}{x} $$
$$y' = \frac{1}{n} x^{\frac{1}{n}}{x^{(-1)}} $$
$$y' = \frac{1}{n} x^{\frac{1}{n} - 1 }$$