Why is gradient describe the steepest ascent direction and not the steepest descent

Question

I have checked all over the internet and I cannot find why is gradient shows you the steepest ascent and not the steepest descent

How can we proof that?

Answer 1

The directional derivative of function $f$ at point $a$ along vector $\bf v$ is ${\bf v} \cdot {\nabla f(a)}$. So if ${\bf v} = \nabla f(a)$, we get $(\nabla f(a)) \cdot (\nabla f(a))$. But the dot product of any vector with itself $\ge 0$. Therefore it's ascent, not descent.