## Leibnitz Theorem

**Leibnitz Theorem:**

We know that the product of two n times differentiable functions is n times differentiable as well. The Leibnitz formula gives an explicit expression for the n-th derivative of the product:

Where as usual is the binomial coefficients " n choose m".

Proof : We use induction with respect to n. For n = 1 the formula is correct. Suppose it is correct for the n-th derivatives and check its correctness for the n 1-st derivatives:

Compliting the arguments.

**Find the n ^{th }derivatives of e^{ax }cos(bx c)**

**Solutions: ** by product rule.

Where we have used the formula Cos A Cos B - Sin A Sin B = Cos ( A B ). Differentiate again and simplifying as before.