# Difficulty Understanding Implicit Differentiation

I am struggling with an Implicit Differentiation question which is as follows:

$z = (7x^4)*\ln(x)4$ where $z$ and $x$ are functions of $t$. $\frac{dx}{dt} = 4$ when $x = e$. Calculate $\frac{dz}{dt}$.

What I have tried so far is finding $\frac{dz}{dx}$ which I believe is $(7x^3) + (28x^3)*\ln(x)$ which I have used the chain rule to get this answer for $\frac{dz}{dx}.$

This is the part where I have tried several things such as multiplying $\frac{dz}{dx}$ by $\frac{dx}{dt}$ but I'm not sure how that works and I have tried replacing the $x$'s with $e$'s and equation $\frac{dx}{dt}$ to $\frac{dz}{dx}$ but I am really unsure what I'm supposed to be doing in this question and nothing appears right.

Any help would be greatly appreciated.

Thank you

you are almost there. what you need is $$\frac{dz}{dt} = \frac{dz}{dx}\frac{dx}{dt} = \left(7x^3+28x^3\ln x\right)\big|_{x = e} \frac{dx}{dt} = (7e^3 + 28e^3)4 = 140e^3$$