The total derivative (ou derivative in the au sense of Fréchet) exists and equals the gradient of f of a (${\displaystyle \mathbf {\nabla f(a)} }$) if ${\displaystyle \lim _{h\to 0}{{f(\mathbf {a+h} )-f(\mathbf {a} )-\langle \mathbf {\nabla f(\mathbf {a} )} ,\mathbf {h} \rangle } \over \|\mathbf {h} \|}=0}$.

Idea

${\displaystyle f(\mathbf {a+h} )=f(\mathbf {a} )+\langle \mathbf {\nabla f(\mathbf {a} )} ,\mathbf {h} \rangle +r(\mathbf {h} )}$

As example R², ${\displaystyle f(a+h)}$, where h is a very small number, equals the sum:

• ${\displaystyle f(a)}$
• ${\displaystyle g(h)}$ where ${\displaystyle g(x)={\frac {df}{dx}}(a)*x}$ (straight line with gradient of the function at the point ${\displaystyle (a,f(a))\,}$)
• the rest ${\displaystyle r(h)}$ which depends only of h

So ${\displaystyle r(\mathbf {h} )=f(\mathbf {a+h} )-f(\mathbf {a} )-\langle \mathbf {\nabla f(\mathbf {a} )} ,\mathbf {h} \rangle }$ ^[1]

See also

• Gradient
• Fréchet derivative