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 nth derivatives of eax 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.