Question

Use the chain rule to find ∂z/∂s and ∂z/∂t. z = sin(θ) cos(ϕ), θ = st5, ϕ = s7t

Answer 1

$z=\sin\theta\cos\phi$
$\theta=st^5$
$\phi=s^7t$

$\dfrac{\partial z}{\partial s}=\dfrac{\partial z}{\partial\theta}\dfrac{\partial\theta}{\partial s}+\dfrac{\partial z}{\partial\phi}\dfrac{\partial\phi}{\partial s}$
$\dfrac{\partial z}{\partial s}=\cos\theta\cos\phi t^5-\sin\theta\sin\phi (7s^6t)$
$\dfrac{\partial z}{\partial s}=t^5\cos(st^5)\cos(s^7t)-7s^6t\sin(st^5)\sin(s^7t)$

$\dfrac{\partial z}{\partial t}=\dfrac{\partial z}{\partial\theta}\dfrac{\partial\theta}{\partial t}+\dfrac{\partial z}{\partial\phi}\dfrac{\partial\phi}{\partial t}$
$\dfrac{\partial z}{\partial t}=\cos\theta\cos\phi(s5t^4)-\sin\theta\sin\phi s^7$
$\dfrac{\partial z}{\partial t}=5st^4\cos(st^5)\cos(s^7t)-s^7\sin(st^5)\sin(s^7t)$