Suppose y=r²x-rx³+5r . g is the SAME as the the derivative of y with respect to r. Find dg/dx
To find dg/dx, we need to differentiate g with respect to x. In order to do this, we can first differentiate y with respect to r, and then differentiate the resulting expression with respect to x.
Differentiating y with respect to r using the power rule gives:
dy/dr = 2rx - 3rx² + 5
Now we can differentiate dy/dr with respect to x:
(d²y/dr²)(dr/dx) = 2r - 6rx
Since g is the same as dy/dr, we can substitute it into the expression above:
dg/dx = 2r - 6rx
So, dg/dx = 2r - 6rx.
To find dg/dx, we need to find the derivative of g with respect to x. To do this, we first need to find the derivative of y with respect to r.
Given y = r²x - rx³ + 5r, let's find dy/dx (the derivative of y with respect to r) by treating x as a constant:
dy/dr = 2rx - 3rx² + 5
Now, let's find dg/dx (the derivative of g with respect to x):
dg/dx = (dy/dr) * (dr/dx)
To find dr/dx, we need to use the chain rule. Since g is the derivative of y with respect to r, it means that g = dy/dr. Therefore, dg/dx can be written as:
dg/dx = (dy/dr) * (dr/dr) * (dr/dx)
Since dr/dr is always equal to 1, we can simplify the equation further:
dg/dx = (dy/dr) * (dr/dx)
Now, substituting the previously calculated dy/dr:
dg/dx = (2rx - 3rx² + 5) * (dr/dx)
Hence, dg/dx = (2rx - 3rx² + 5) * (dr/dx)