Calculus, The Derivative of xr

The Derivative of xr

Let r be a rational exponent k/l. We would like to differentiate the function xr, at least for x > 0. We already know the derivative of x to the r when r is an integer.

Note that y = x1/l and x = yl are inverse functions, and the derivative of the latter is lyl-1. Apply the inverse function rule, and the derivative of the former is 1 over lyl-1. Substitute y = x1/l and simplify, giving 1/l times x1/l-1. This is the same formula we saw when the exponent was an integer. The exponent is folded into the coefficient, and is then decreased by 1.

For instance, the cube root of x, which is x to the 1/3 power, has a derivative of 1/3 x-2/3. This can also be written as 1 over 3x2/3.

Next, write xk/l as x1/l raised to the k. Use the chain rule and simplify the result to obtain rxr-1.

Finally r might be negative. If it is, set r = -s and differentiate 1/xs, using the quotient rule. Again, the answer is rxr-1.

If you're comfortable with limits and continuity, you can prove that this formula holds for any real exponent r. The integral of the limit of a uniformly convergent sequence of functions is the limit of the integrals, and we can build a sequence of rational power functions that approach rxr-1. The integral is xr, and by the fundamental theorem of calculus, the derivative of xr is rxr-1.

If we want to know the derivative of xπ, and I'm not sure why we would, it is πxπ-1.

Assume a steady human, or a pump, pushes air into a balloon at 1 liter per minute. The diameter of the balloon is proportional to the cube root of its volume. Don't worry about the constant of proportionality for now. Differentiate this to find out how quickly the diameter is increasing. If t is time and k is the constant, the diameter is kt1/3, so the derivative is (k/3)t-2/3. The derivative gets smaller and smaller with time, because the diameter's expansion proceeds more slowly as the balloon grows larger. Yet the derivative is always positive; the diameter is always increasing.