Question

Let w=2 x²+y³ ; x=4 s+t, y=5 s-2 t. Find ∂ w/∂ t and ∂ w/∂ s

Answer 1

To find the partial derivatives of w with respect to t and s, substitute the values of x and y into the equation for w, and differentiate with respect to t and s respectively.

Step-by-step explanation:

To find the partial derivatives of w with respect to t and s, we need to substitute the values of x and y into the equation for w, and differentiate with respect to t and s respectively:

First, substitute x=4s+t and y=5s-2t into w=2x²+y³:

w = 2(4s+t)² + (5s-2t)³

= 32s² + 16st + 2t² + 125s³ - 100s²t + 20st² - 8t³

To find ∂ w/∂ t, differentiate w with respect to t, treating s as a constant:

∂ w/∂ t = 16s + 4t - 24t²

To find ∂ w/∂ s, differentiate w with respect to s, treating t as a constant:

∂ w/∂ s = 64s + 16t + 375s² - 200st + 20t²