"If y is a differentialble function of u, u is a diffferentiable function of v, and v is a differentiable function of x, then dy/dx=dy/du *du/dv *dy/dx
The given expression for dy/dx is incorrect. The correct expression should be:
dy/dx = (dy/du) * (du/dv) * (dv/dx)
To understand why this is the correct expression, let's first break it down step by step:
1. We have a function y that is differentiable with respect to u. This means that dy/du exists.
2. We also have a function u that is differentiable with respect to v. This means that du/dv exists.
3. Finally, we have a function v that is differentiable with respect to x. This means that dv/dx exists.
Now, to find the derivative of y with respect to x (dy/dx), we apply the chain rule of differentiation. According to the chain rule, if we have a composition of functions, the derivative of the composition with respect to x is the product of the derivatives of the individual functions with respect to their respective variables.
In this case, we have y as a composition of u, v, and x. So, by applying the chain rule, we get:
dy/dx = (dy/du) * (du/dv) * (dv/dx)
You can think of it as taking the derivative of y with respect to u, then multiplying it by the derivative of u with respect to v, and finally multiplying it by the derivative of v with respect to x.
This expression accounts for the chain of differentiations happening from y to x through u and v, which is why it correctly represents the derivative of y with respect to x.