The Power Rule d r What is the derivative of x ? We answered this question ﬁrst for positive dx integer values of r, for all integers, and then for rational values of r: d r x = rxr−1 dx We’ll now prove that this is true for any real number r. We can do this two ways: 1st method: base e Since x = eln x , we can say: eln x �r xr = � xr = er ln x We take the derivative of both sides to get: d r x dx = = = d r x dx = d r ln x d e = er ln x (r ln x) (by the chain rule) dx � � dx r er ln x (remember r is constant) � r �x (because xr = er ln x ) xr x rxr−1 2nd method: logarithmic diﬀerentiation We deﬁne f (x) = xr , and take the natural log of both sides to get ln f = r ln x. The technique of logarithmic diﬀerentiation requires us to we plug our function into the formula: (ln f )� = f� f So we ﬁrst compute: ln f = ln xr ln f = r ln x And then take the derivative of both sides to get: r (ln f )� = x 1 Since (ln f )� = f� f , we have: f � = f (ln f )� xr = �r� x = rxr−1 . Look over the two methods again – the calculations are almost the same. This is typical. To use the second method we had to introduce a new symbol like u or f . In the ﬁrst method we had to deal with exponents. It’s worthwhile know both methods. 2