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

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

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

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

and partially differentiate H: Are the following differentials correct?

$\displaystyle \frac{\partial H}{\partial c(t)} = 1 + \phi'(c(t))\frac{1-\Psi(t)}{2} \frac{\phi(t)}{(\phi(c(t)))^2}$

$\displaystyle \frac{\partial H}{\partial t_{EC}} = -1$

$\displaystyle \frac{\partial H}{\partial t_{WA_c}} = \frac{1-\Psi(t_{WA_c})}{2} \frac{\phi(t_{WA_c})}{\phi(c(t_{WA_c}))}$

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

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

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

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

or

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

What is correct???