Question

Question #b8c9a

Answer 1

Let `y=x^{p/q}`, where `p` and `q` are nonzero integers.

By raising to the qth-power,

`Rightarrow y^q=x^p`

by implicitly differentiating with respect to `x`,

`Rightarrow qy^{q-1}cdot{dy}/{dx}=px^{p-1}`

by dividing by `qy^{q-1}=q(x^{p/q})^{q-1}=qx^{p-p/q}`,

`Rightarrow{dy}/{dx}={px^{p-1}}/{qx^{p-p/q}}={p}/{q}x^{p/q-1}`.

Hence, the power rule works for rational powers.

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I hope that this was helpful.