The problem asks to find $\frac{\partial w}{\partial t}$ using the chain rule, where $w = \sqrt{x^2 + y^2 + z^2}$, $x = \cos(st)$, $y = \sin(st)$, and $z = s^2 t$. The final answer should be expressed in terms of $s$ and $t$.
The problem asks to find ∂t∂w using the chain rule, where w=x2+y2+z2, x=cos(st), y=sin(st), and z=s2t. The final answer should be expressed in terms of s and t.
2. Solution Steps
First, write down the chain rule for ∂t∂w:
∂t∂w=∂x∂w∂t∂x+∂y∂w∂t∂y+∂z∂w∂t∂z
Next, compute the partial derivatives:
∂x∂w=2x2+y2+z21⋅2x=x2+y2+z2x
∂y∂w=2x2+y2+z21⋅2y=x2+y2+z2y
∂z∂w=2x2+y2+z21⋅2z=x2+y2+z2z
∂t∂x=−ssin(st)
∂t∂y=scos(st)
∂t∂z=s2
Now, substitute these into the chain rule formula:
