Answer
a. $4t \tan^{-1} t+1$ ;
b. $\pi+1$
Work Step by Step
a. Using chain rule.
$\dfrac{dw}{dt}=\dfrac{\partial w}{\partial x}\dfrac{dx}{dt}+\dfrac{\partial w}{\partial y}\dfrac{dy}{dt}+\dfrac{\partial w}{\partial z}\dfrac{dz}{dt}$
or, $\dfrac{4yte^x}{(t^2+1)}+\dfrac{2e^x}{(t^2+1)}-(\dfrac{e^t}{z})=(4t) \tan^{-1} t+1$
Now, by using direct differentiation.
we have, $w=2ye^x-\ln z$
Also, $w=(2 \tan^{-1} t) (t^1+1)-t$
and $\dfrac{dw}{dt}=(4t) \tan^{-1} t+1$
b. $\dfrac{dw}{dt}(1)=4t \tan^{-1} t+1=4(1) \tan^{-1} (1)+1=4(1) (\dfrac{\pi}{4})+1=\pi+1$
