# Log Derivative Trick

A very simple property from calculus appears frequently in machine
learning, statistics, reinforcement learning and other fields. It is the statement
that for any function \(f(x)\), the following equality holds:

\[\nabla_x f(x) = f(x) \nabla_x \log f(x)\]
## Derivation

The chain rule tells us that for any function \(f(x)\):

\[\nabla_x \log f(x) = \frac{1}{f(x)} \nabla_x f(x)\]
Rearranging, we obtain the desired equality:

\[f(x) \nabla_x \log f(x) = \nabla_x f(x)\]
## Applications

### Policy Gradient Theorem

In reinforcement learning, policy-based RL relies on the
policy-gradient theorem,
which uses the log derivative trick.