Calculate the derivative of a data-function in r

Question

Is there an easy way to calculate the derivative of non-liner functions that are give by data?

for example:

x = 1 / c(1000:1)

y = x^-1.5
ycs = cumsum(y)

plot (x, ycs, log="xy")

How can I calculate the derivative function from the function given by ´x´ and ´ycs´?

Answer 1

Was also going to suggest an example of a smoothed spline fit followed by prediction of the derivative. In this case, the results are very similar to the diff calculation described by @dbaupp:

spl <- smooth.spline(x, y=ycs)
pred <- predict(spl)

plot (x, ycs, log="xy")
lines(pred, col=2)

ycs.prime <- diff(ycs)/diff(x)
pred.prime <- predict(spl, deriv=1)

plot(ycs.prime)
lines(pred.prime$y, col=2)

Answer 2

Generating derivatives from raw data is risky unless you are very careful. Not for nothing is this process known as "error multiplier." Unless you know the noise content of your data and take some action (e.g. spline) to remove the noise prior to differentiation, you may well end up with a scary curve indeed.