Chain Rule — Question 1

Question 1 – Solution

Strategy. Combine the two routes $t\to x\to z$ and $t\to y\to z$.

Step 1: Ingredients $$z_x=2xy,\qquad z_y=x^2+\cos y,\qquad x'=2t,\qquad y'=e^t.$$ Therefore $$\boxed{\frac{dz}{dt}=2xy(2t)+(x^2+\cos y)e^t}.$$

Step 2: Evaluate At $t=0$, $x=-1$ and $y=1$, giving $$\boxed{\left.\frac{dz}{dt}\right|_{t=0}=1+\cos 1}.$$

Step 3: Direct check The composite is $$z(t)=(t^2-1)^2e^t+\sin(e^t).$$ Its derivative is $4t(t^2-1)e^t+(t^2-1)^2e^t+\cos(e^t)e^t$, which at zero is $1+\cos 1$.