Hi! I am trying to study for an upcoming exam by doing online problems, but this one has me completely stumped. If someone can help answer or explain how to do this problem I would really appreciate it.
2026-04-03 10:47:00.1775213220
The chain rule: differentiation in several variables
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Take the partial derivative of $f$ with respect to $s$ as follows
$$\begin{align} \frac{\partial}{\partial s}f(x,y) &= \frac{\partial f}{\partial x}\frac{\partial x}{\partial s} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial s} \\&= 7\cos(y)\frac{\partial x}{\partial s} - 7x\sin(y)\frac{\partial y}{\partial s} \\&= 7(8s^7)\cos(y) + 77x\sin(y) \\&= 56s^7\cos(11r - 11s) + 77(r^8 + s^8)\sin(11r - 11s) \,\,. \end{align}$$ Now substitute the point $(r,s) = (2,1)$ to complete the calculation.