Can someone show me how to prove the following? I have $f(x)=g(ax+b)$, a and b are constant.
I need to show that $\nabla f(x)=a\nabla g(x)$ and $\nabla^2 f(x)=a^2\nabla^2 g(x)$...
I was thinking that the final answer should have ax+b in it, but apparently it can be shown that the above is true???  
 A: In this problem $f(x) = g(h(x))$, where $h(x) = ax + b$.  I'm going to consider the case where $a$ is a matrix rather than a scalar, because it's useful and no more difficult.  You can assume $a$ is a scalar if you'd like.
Let's establish some notation.  Recall that if $F:\mathbb R^n \to \mathbb R^m$ is differentiable at $x$, then $F'(x)$ is an $m \times n$ matrix.  In the special case where $m = 1$, $F'(x)$ is a $1 \times n$ matrix.  I'm going to use the convention that $\nabla F(x) = F'(x)^T$, so $\nabla F(x)$ is a column vector rather than a row vector.  Then $G(x) = \nabla F(x)$ is a function from $\mathbb R^n \to \mathbb R^n$, and $\nabla^2 F(x) = G'(x)$, which is an $n \times n$ matrix.
The chain rule tells us that
\begin{align}
f'(x) &= g'(h(x))h'(x) \\
&= g'(ax + b) a.
\end{align}
It follows that
\begin{align}
\nabla f(x) &= a^T g'(ax+b)^T \\
&= a^T \nabla g(ax + b).
\end{align}
That is our formula for $\nabla f(x)$.
Preparing to use the chain rule again, we can express $\nabla f(x)$ as
$\nabla f(x) = w(h(x))$, where
$w(x) = a^T \nabla g(x)$.
Note that $w'(x) = a^T \nabla^2 g(x)$.
Applying the chain rule to $z(x) = \nabla f(x) = w(h(x))$, we see that
\begin{align}
\nabla^2 f(x) &= w'(h(x))h'(x) \\
&= a^T \nabla^2 g(ax + b) a.
\end{align}
This is our formula for $\nabla^2 f(x)$.
