Suppose I define a function $f(x)$ $$f(x)=\sqrt{1+{\left(\frac{\partial u}{\partial x}\right)}^2}$$ and $$u(x)=\sum_{k=0}^{\infty}a_k x^k\quad\forall k\in\mathbb{Z}$$ Now, if I want to find out $\frac{\partial f}{\partial a_m}$ where $m\in[0,\infty)$, then I can do it in two ways $$\frac{\partial f}{\partial a_m}=\frac{\partial f}{\partial {u}'}.\frac{\partial u'}{\partial a_m}=\frac{1}{2\sqrt{1+{\frac{\partial u}{\partial x}}^2}}.2u'.\frac{\partial u'}{\partial a_m}=\frac{\sum_{k=0}^{\infty}ka_kx^{k-1}.mx^{m-1}}{\sqrt{1+{(\sum_{k=0}^{\infty}ka_kx^{k-1})}^2}}$$ or $$\frac{\partial f}{\partial a_m}=\frac{\partial f}{\partial u'}.\frac{\partial u'}{\partial x}.\frac{1}{u'}.\frac{\partial u}{\partial a_m}=\frac{\sum_{k=0}^{\infty}k(k-1)x^{k-2}}{\sqrt{1+(\sum_{k=0}^{\infty}ka_kx^{k-1})^2}}.x^m$$
Why do I get different results? Where am I going wrong?