d/dx(lnu)
u’/u
To find the derivative of ln(u) with respect to x, we’ll need to use the chain rule.
Let’s begin by defining a new function v = u(x). Then we have
y = ln(v)
Now we can use the chain rule, which says that
dy/dx = dy/dv * dv/dx
The first factor is easy to find, since the derivative of ln(v) with respect to v is simply 1/v:
dy/dv = 1/v
The second factor is also straightforward, since we defined v = u(x):
dv/dx = du/dx
Putting it all together, we have:
dy/dx = dy/dv * dv/dx
= (1/v) * (du/dx)
= (1/u(x)) * (du/dx)
So finally we can write:
d/dx (ln(u)) = (1/u) * (du/dx)
Thus, the derivative of ln(u) with respect to x is (1/u) * (du/dx).
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