Hello MHF,
Would appreciate some help with the following question,
Find $\displaystyle \frac{dz}{dt}$ if $\displaystyle z=txy^2 ,\ x=t+\ln(y+t^2)\ \mbox{and}\ y=e^2$. You may express your answer as a function of $\displaystyle t,x,y.$
Hello MHF,
Would appreciate some help with the following question,
Find $\displaystyle \frac{dz}{dt}$ if $\displaystyle z=txy^2 ,\ x=t+\ln(y+t^2)\ \mbox{and}\ y=e^2$. You may express your answer as a function of $\displaystyle t,x,y.$
Actually, here, your original function z is a function of three variables, x, y, and t.
The chain rule for this would be $\displaystyle \frac{dz}{dt}= \frac{\partial z}{\partial t}+ \frac{\partial z}{\partial x}\frac{dx}{dt}+ \frac{\partial z}{\partial y}\frac{dy}{dt}$
The $\displaystyle \frac{\partial z}{\partial t}$, of course, would just be $\displaystyle xy^2$.