Evaluate using Leibniz rule and/or chain rule

[Alex_ra]

I want to evaluate
$$ \frac{d}{dt}\int_{0}^{^{\eta(t)}}\rho(p,t)dz $$

where p itself is $$ p=p(z,t) $$

I have the feeling I have to use Leibniz rule and/or chain rule, but I'm not sure how...

Thanks.

 

[fresh_42]

I want to evaluate
$$ \frac{d}{dt}\int_{0}^{^{\eta(t)}}\rho(p,t)dz $$

where p itself is $$ p=p(z,t) $$

I have the feeling I have to use Leibniz rule and/or chain rule, but I'm not sure how...

Thanks.

So $\rho(p,t)=\rho(p(z,t),t)=\varphi(z,t)$, and if $\phi_z(z,t)$ is the anti-derivative of $\varphi(z,t)$ with respect to the first coordinate, your are actually looking for $\frac{d}{dt} \left( \phi_z(\eta(t),t) - \phi_z(0,t) \right)$.
Don't know whether this helps.

 

[lurflurf]

The chain rule let's us differentiate each instance of the variable separately.
$$\dfrac{d}{dt}\int_{0}^{^{\eta(t)}}\rho(p(z,t),t)dz=\left.\left(\dfrac{\partial}{\partial u}+\dfrac{\partial}{\partial v}+\dfrac{\partial}{\partial w}\right)\int_{0}^{^{\eta(u)}}\rho(p(z,v),w)dz\right|_{u,v,w=t}$$
So replace each t by a dummy variable differentiate with respect to each and add them up, then replace each with t again.

 

[Alex_ra]

Thanks fresh_24 and lurflurf for your quick and helfpul answers! Have a nice day.

 

[lavinia]

$$ F(t) = \int_{0}^{^{t}}\rho(p,t)dz $$ is a function of $t$.

Your function is $F(η(t))$.