Taylor series and differentiation

Question

The Taylor series about 0 for the function $$f(x) = \left(\frac14 + x\right) ^ {-\frac{3}{2}}$$ is

$$f(x) = 8 - 48x+240x^2-1120x^3 + \dots$$

for $-\frac{1}4 < x <\frac14$

Use differentiation to find the Taylor series about $0$ for the function

$$g(x) = \left(\frac14+x\right)^{-\frac{5}2}$$ giving all terms up and including the one in $x^2$.

Answer 1

Hint: take a look at $f'(x)$ and how it is connected to $g$.