# Nth Derivative of a Function

1. Aug 26, 2013

### FeDeX_LaTeX

1. The problem statement, all variables and given/known data
Let $f(x) = \frac{\sin x}{b + \cos(ax)}$. Show that the nth derivative $f^{(n)}(0) = 0$ if n is an even integer.

2. Relevant equations
Leibniz's generalised product rule:

$(f \cdot g)^{(n)} = \sum_{k = 0}^{n} \binom{n}{k} f^{(k)}g^{(n-k)}$

3. The attempt at a solution
I'm letting $f(x) = \sin x$ and $g(x) = \frac{1}{b + \cos(ax)}$ then applying Leibniz's rule. Clearly, the terms of the series k = 0, k = 2, ... (every even k) are all 0 when x = 0, since they all contain an even derivative of sin (which gives us sin again). But what do I do about the derivatives of g(x)? Is this the right approach?

2. Aug 26, 2013

### Mathitalian

Odd functions of x have only odd powers of x in their Taylor-McLaurin series, so...

3. Aug 27, 2013

### FeDeX_LaTeX

So you're saying I let $\frac{1}{b + \cos(ax)} = a_0 + a_{1}x^2 + a_{2}x^4 + ...$?

4. Aug 27, 2013

### Ray Vickson

There is a much, much easier way. Just answer the following three questions.
1. The given function is (a) odd; (b) even; (c) neither.
2. The derivative of an even function is (a) even; (b) odd; (c) neither.
3. The derivative of an odd function is (a) even; (b) odd; (c) neither,