Question

Given y=f(u) and u=g(x) , find dy/dx by using Leibniz's notation for the chain rule: dy/dx = dy/du du/dx .

y=5u⁴+2 u=3x³

Answer 1

To find dy/dx, we need to find dy/du and du/dx and then apply the chain rule. Given y=f(u) and u=g(x), we have y=5u^4+2 and u=3x^3. First, differentiate y with respect to u, then differentiate u with respect to x, and finally apply the chain rule to find dy/dx.

Step-by-step explanation:

To find ∂y/∂x, we need to find ∂y/∂u and ∂u/∂x and then apply the chain rule.

Given y = f(u) and u = g(x), we have y = 5u⁴ + 2 and u = 3x⁴.

First, we differentiate y with respect to u: ∂y/∂u = 20u⁳.

Next, we differentiate u with respect to x: ∂u/∂x = 9x⁳.

Finally, applying the chain rule, we have ∂y/∂x = (∂y/∂u)(∂u/∂x) = (20u⁳)(9x⁳) = 180u⁳x⁳ = 180(3x⁳)(x⁳) = 540x⁵