I have some problems with a differential - don't now what is correct...

Start with c(t) = t_{EC} + \int_{t_{WA_c}}^{t} \frac{1-\Psi(s)}{2} \frac{\phi(s)}{\phi(c(s))}ds.

If I want to use the implicit function theorem to differentiate get \frac{d c(t)}{d t_{EC}} I define

H \equiv c(t) - t_{EC} - \int_{t_{WA_c}}^{t} \frac{1-\Psi(s)}{2} \frac{\phi(s)}{\phi(c(s))}ds

and partially differentiate H: Are the following differentials correct?

\frac{\partial H}{\partial c(t)} = 1 +  \phi'(c(t))\frac{1-\Psi(t)}{2} \frac{\phi(t)}{(\phi(c(t)))^2}
\frac{\partial H}{\partial t_{EC}} = -1
\frac{\partial H}{\partial t_{WA_c}} = \frac{1-\Psi(t_{WA_c})}{2} \frac{\phi(t_{WA_c})}{\phi(c(t_{WA_c}))}

or is the first one wrong??? Should it be instead

\frac{\partial H}{\partial c(t)} = 1 + \int_{t_{WA_c}}^{t} \phi'(c(s))\frac{1-\Psi(s)}{2} \frac{\phi(s)}{(\phi(c(s)))^2}ds

Or completely different??? Because then I can use the implicit function theorem to get


\frac{d c(t)}{d t_{EC}} = \frac{1 - \frac{1-\Psi(t_{WA_c})}{2} \frac{\phi(t_{WA_c})}{\phi(c(t_{WA_c}))} \frac{d t_{WA_c}}{d t_{EC}}}{1 +  \phi'(c(t))\frac{1-\Psi(t)}{2} \frac{\phi(t)}{(\phi(c(t)))^2}}

or

\frac{d c(t)}{d t_{EC}} = \frac{1 - \frac{1-\Psi(t_{WA_c})}{2} \frac{\phi(t_{WA_c})}{\phi(c(t_{WA_c}))} \frac{d t_{WA_c}}{d t_{EC}}}{1 + \int_{t_{WA_c}}^{t} \phi'(c(s))\frac{1-\Psi(s)}{2} \frac{\phi(s)}{(\phi(c(s)))^2}ds}

What is correct???