Derivatives of rational powers of x

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



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

Next: Derivatives of Logarithmic Functions