## Example 8.4.18 (c): Finding Taylor Series by Differentiation

Start with a known series and differentiate both sides.

### Example

Find a series for*f(x) =*^{2x}/_{(1-x2)2}This time we need to find a function whose derivative is our function in question, and whose Taylor series we know. Experimenting on scratch paper for a little while will give us, as a guess, the function

g(x) =^{1}/_{1-x2}

The derivative of this function is
*g'(x) = ^{2x}/_{(1-x2)2}*, our
function in question. Thus:

f(x) = g'(x) = g(x) = x^{2n}= x^{2n}= 2n x^{2n-1}

for *|x| < 1*. We have, of course, quietly used substitution
for the Geometric series and as a result - voila! - we have a Taylor series
centered at zero for our function, just as desired.

2n x^{2n-1}f(x) =^{2x}/_{(1-x2)2}