d/dx ln(x)
To find the derivative of ln(x) with respect to x, we can use the chain rule
To find the derivative of ln(x) with respect to x, we can use the chain rule. The chain rule states that if we have a composite function f(g(x)), then the derivative of f(g(x)) with respect to x is given by f'(g(x)) * g'(x).
In this case, ln(x) can be thought of as the composition of two functions: the natural logarithm function f(u) = ln(u), and the identity function g(x) = x.
Using the chain rule, we have:
d/dx ln(x) = d/dx (ln(g(x))) = f'(g(x)) * g'(x)
Now, let’s find the derivatives of the two individual functions:
f'(u) = d/du (ln(u)) = 1/u (this is a well-known property of the natural logarithm)
g'(x) = d/dx(x) = 1 (derivative of x with respect to x is 1)
Now we can substitute these derivatives into the chain rule formula:
d/dx ln(x) = f'(g(x)) * g'(x) = 1/g(x) * 1 = 1/x
Therefore, the derivative of ln(x) with respect to x is 1/x.
In summary, d/dx ln(x) = 1/x
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