\begin{eqnarray*} y & = & f(t) \\ t & = & g(x) \\ y & =& f(g(x)) \\ \frac {dy}{dx} & = & \frac {dy}{dt} * \frac {dt}{dx} \\ & & \frac {dy}{dt} = f'(t) = f'(g(x)) \\ & & \because{ \frac {dt}{dx} = g'(x) } \\ & & \frac {dy}{dx} = f'(g(x)) * g'(x) \\ \end{eqnarray*}

\begin{eqnarray*} y & = & (2x^2 + 1)^2 \\ t & = & 2x^2 + 1 \\ y & = & t^2 \\ t & = & 2x^2 + 1 \\ y' & = & 2t \\ t' & = & 4x \\ \frac{dy}{dx} & = & \frac{dy}{dt} * \frac{dt}{dx} \\ \frac{dy}{dx} & = & 2(t^{2-1}} * 4 x^{2-1} \\ & = & 2(2x^2 + 1) * 4x \\ & = & (4x^2 + 2) * 4x \\ & = & 16x^3 + 8x \\ - y & =& f(g(x)) \\ \frac {dy}{dx} & = & \frac {dy}{dt} * \frac {dt}{dx} \\ & & \frac {dy}{dt} = f'(t) = f'(g(x)) \\ & & \because{ \frac {dt}{dx} = g'(x) } \\ & & \frac {dy}{dx} = f'(g(x)) * g'(x) \\ \end{eqnarray*}