From the following equation, we show that the formula

is valid for all integer values of *no* is for *no* = . We will now use implicit differentiation to show that this formula is valid for any rational exponent. More precisely, we will show that if *r* is a rational number then

whenever and are set. For now, we will admit, without proof that is differentiable.

Be *y* = . Once *r* is a rational number, can be expressed as an integer ratio *r = m / n.* Like this,* y* = = can be written as

Implicitly differentiating with respect to *x* and using we get

This way it can be written as

**Example**

From

If *u* is a differentiable function of *x* and* r* is a rational number, then the chain rule gives rise to the following generalization of