First Variation of a Functional: Frechet derivative I am trying to solve the following question.
(First Variation of a Functional) Consider the normed linear space $C^1[0, 1]$ with the norm
$$\| u \| = \max_{0\leq t \leq 1} |u(t)| + \max_{0\leq t \leq 1} |u'(t)| ,  \forall u \in C^1[0, 1].$$
Consider the functional 
$$J : C^1[0, 1] \rightarrow R, \ J(u) = \int_{0}^1 f(t, u(t), u'(t)) dt,$$
where f is a $C^1$ function. Show that the Frechet derivative can be written in the form:
$$J'(u)h=\int_{0}^1[f_u(t,u(t),u'(t))h(t) + f_{u'}(t,u(t),u'(t))h']dt $$
Hint: Use the Taylor series expansion for $f$:
$$f(t, u(t) + h(t), u' (t) + h'(t)) = f(t,u,u') + f_u(t,c(t),u'(t))h(t) + f_u(t,u(t),d(t))h'(t)$$
where $c(t)$ lies between $u(t)$ and $u(t)+h(t)$ and $d(t)$ lies between $u'(t)$ and $u'(t)+h'(t)$.
Here's what I've tried. We have $$J(u+h) = \int_{0}^1 f(t,u(t)+h(t),u'(t)+h'(t))dt$$
$$ = \int_{0}^1 [ f(t,u,u') + f_u(t,c(t),u'(t))h(t) + f_u(t,u(t),d(t))h'(t)]dt$$
$$ = J(u) + \int_{0}^1 [ f_u(t,c(t),u'(t))h(t) + f_u(t,u(t),d(t))h'(t)]dt $$
It now suffices to show that
$$ \lim_{h\rightarrow 0} \frac{ \|  \int_{0}^1 [ f_u(t,c(t),u'(t))h(t) + f_u(t,u(t),d(t))h'(t)]dt - J'(u)h  \| }{\|h\|}=0$$ 
Substitute $J'(u)h$ into the last equation and simplify the LHS:
$$ \lim_{h\rightarrow 0} \frac{ \|  \int_{0}^1 [ f_u(t,c(t),u'(t))h(t) + f_u(t,u(t),d(t))h'(t)]dt - \int_{0}^1[f_u(t,u(t),u'(t))h(t) + f_{u'}(t,u(t),u'(t))h']dt  \| }{\|h\|}$$ 
$$= \lim_{h\rightarrow 0} \frac{ \|  \int_{0}^1 [ \bigg( f_u(t,c(t),u'(t)) -f_u(t,u(t),u'(t))\bigg) h(t) + \bigg(f_u(t,u(t),d(t))-f_{u'}(t,u(t),u'(t))\bigg)h'(t)]dt  }{\|h\|}$$
Use Triangle inequality and Squeeze theorem:
$$\leq \lim_{h\rightarrow 0} \frac{ \|  \int_{0}^1 [ \bigg( f_u(t,c(t),u'(t)) -f_u(t,u(t),u'(t))\bigg) h(t)] dt \|}{\|h\|} + \frac{ \|  \int_{0}^1 [ \bigg(f_u(t,u(t),d(t))-f_{u'}(t,u(t),u'(t))\bigg)h'(t)]dt  }{\|h\|} $$
$$ \leq \lim_{h\rightarrow 0} \frac{ \| h\|_\infty | \int_{0}^1   f_u(t,c(t),u'(t)) -f_u(t,u(t),u'(t))   dt | }{\|h\|} 
+ \frac{  \| h' \|_\infty  | \int_{0}^1  f_u(t,u(t),d(t))-f_{u'}(t,u(t),u'(t)) dt|  }{\|h\|}$$
Since $\| h\| = \|h\|_\infty + \| h'\|_\infty$, we have $\| h\| \geq \|h\|_\infty $ and $\| h\| \geq  \| h'\|_\infty$. So,
$$ \leq \lim_{h\rightarrow 0} | \int_{0}^1   f_u(t,c(t),u'(t)) -f_u(t,u(t),u'(t))   dt |
+  | \int_{0}^1   f_u(t,u(t),d(t))-f_{u'}(t,u(t),u'(t))  dt |  $$
(There must be a typo in the question! $f_u(t,u(t),d(t))$ should be $f_{u'}(t,u(t),d(t))$).
$$ \leq \lim_{h\rightarrow 0}  |J_u(c) - J_u(u)| +  | \int_{0}^1   f_{u'}(t,u(t),d(t))-f_{u'}(t,u(t),u'(t))  dt  |$$
As $h$ approaches 0, $c$ approaches $u$. So, 
$$ = 0 +  | \int_{0}^1   f_{u'}(t,u(t),d(t))-f_{u'}(t,u(t),u'(t))  dt  |$$
I'm stuck here.
PS. What does $f_{u'}(t,u(t),u'(t))$ mean?
 A: Hint: if ${\varphi} \colon  \mathbb{R} \rightarrow  \mathbb{R}$ is differentiable in the
interval $\left(x , y\right)$, the following inequality holds by the mean
value theorem
\begin{equation}\left|{\varphi} \left(y\right)-{\varphi} \left(x\right)-{{\varphi}'} \left(x\right) \left(y-x\right)\right|  \leqslant  \left|y-x\right| {\sup }_{z \in  \left(x , y\right)} \left|{{\varphi}'} \left(z\right)-{{\varphi}'} \left(x\right)\right|\end{equation}
Apply this to
\begin{equation}{\varphi} \left({\theta}\right) = f \left(t , u \left(t\right)+{\theta} h \left(t\right) , {u'} \left(t\right)+{\theta} {h'} \left(t\right)\right) \qquad  {\theta} \in  \left[0 , 1\right]\end{equation}
then use the fact that $u \left(t\right) , h \left(t\right) , {u'} \left(t\right) , {h'} \left(t\right)$   stay in
a bounded set of $\mathbb{R}$ and the uniform continuity of the partial
derivatives of $f$ on bounded sets.
Remark: Uniform continuity of the partial derivatives mean that when $B$
is a bounded set of $\mathbb{R}$, there exists a function ${\epsilon} \left(s\right)$
defined on ${\mathbb{R}}^{+}$ such that ${\epsilon} \left(s\right) \mathop{\longrightarrow}\limits_{s \rightarrow  {0}^{+}} 0$
and
\begin{equation}\left|{\partial }_{i} f \left(t , {u}_{1} , {v}_{1}\right)-{\partial }_{i} f \left(t , {u}_{2} , {v}_{2}\right)\right|  \leqslant  {\epsilon} \left(\left|{u}_{1}-{u}_{2}\right|+\left|{v}_{1}-{v}_{2}\right|\right)\end{equation}
whenever $t \in  \left[0 , 1\right]$, ${u}_{1} , {v}_{1} , {u}_{2} , {v}_{2} \in  B$ and ${\partial }_{i} f$ is either ${f}_{u}$ or ${f}_{{u'}}$.
