Final answer:
To use the chain rule to find dw/dt, substitute the given expressions for x, y, and z into the equation for w. Differentiate each term with respect to t and simplify to find dw/dt.
Step-by-step explanation:
To find dw/dt using the chain rule, we need to express w in terms of t. Given that x = t³, y = 4 - t, and z = 9 + 6t, we can substitute these expressions into w = xeʸ/ᶻ.
w = (t³)eʽ(4 - t)ᶻ/(9 + 6t)
Using the chain rule, we differentiate each term with respect to t:
dw/dt = (dx/dt)(eʽ(4 - t)ᶻ/(9 + 6t)) + (t³)(deʽ(4 - t)ᶻ/(9 + 6t))/dt + (t³)eʽ(4 - t)d(ᶻ/(9 + 6t))/dt
We then substitute the given values for dx/dt, de/dt, and dz/dt and simplify to find dw/dt.