Question

Find ∂s/∂w chain rule ∂s/∂w,w=x2+y2+z2,x=5tsin(s),y=5cos(s),z=4st2

Answer 1

To find ∂s/∂w using the chain rule, substitute the given values of x, y, and z into the equation, find ∂s/∂x, ∂s/∂y, ∂s/∂z, and then use the chain rule to evaluate ∂s/∂w. The final result is 0.

Step-by-step explanation:

To find the partial derivative ∂s/∂w using the chain rule, we need to substitute the given values of x, y, and z into the equation. Let's start by finding ∂s/∂x, ∂s/∂y, and ∂s/∂z:

∂s/∂x = 2x = 2(5tsin(s)) = 10tsin(s)

∂s/∂y = 2y = 2(5cos(s)) = 10cos(s)

∂s/∂z = 2z = 2(4st^2) = 8st^2

Now, using the chain rule, we can find ∂s/∂w:

∂s/∂w = (∂s/∂x)(∂x/∂w) + (∂s/∂y)(∂y/∂w) + (∂s/∂z)(∂z/∂w)

Since w is not directly dependent on x, y, or z, their partial derivatives with respect to w are both 0. Therefore, ∂s/∂w = (∂s/∂x)(0) + (∂s/∂y)(0) + (∂s/∂z)(0) = 0.