If is a differentiable function in whose first derivative
is locally integrable, then the derivative defines its own
Integrating by parts and noting that at the boundaries ,
because of compact support, we have
This suggests defining the derivative of any distribution via
We need to check that this definition satisfies the properties of a
distribution. We observe that if is a test function then so is .
Linearity is obvious. What about continuity? If is a zero
sequence, so is the sequence and thus
tends to zero as . Hence (1) defines a
Remark: All distributions are infinitely differentiable is the class
of test functions includes only infinitely differentiable functions.
Comment: One can take test functions which are only differentiable.
Then the distribution will only be times differentiable. Thus, enlarging
the class of test functions reduces the class of distributions.
More generally, if is the th derivative of the distribution ,
Distributions can be generated by functions which are not differentiable in the
ordinary sense. However, we can differentiate them in the
For example, the derivative of the delta distribution gives us the dipole
Also, the derivative of the Heaviside function is given by
Since , from the fundamental theorem of calculus,
Therefore the derivative of the Heaviside function is the delta function.