Derivative of ln x
1/x
The derivative of ln x is given by:
d/dx (ln x) = 1/x
To understand why this is the case, we can use the definition of a derivative.
Let f(x) = ln x. Then, the derivative of f(x) at a point a is defined as:
f'(a) = lim (h->0) ((f(a+h) – f(a))/h)
Substituting ln x for f(x), we get:
f'(a) = lim (h->0) ((ln (a+h) – ln a)/h)
Using the logarithmic identity ln (a/b) = ln a – ln b, we can simplify the above expression to:
f'(a) = lim (h->0) (ln ((a+h)/a))/h)
Using the logarithmic identity ln (1+x) ~ x for small x, we can approximate the above expression as:
f'(a) = lim (h->0) (1/a) ln (1+h/a)
As h->0, h/a->0 and hence, we can further approximate the above expression as:
f'(a) = (1/a) ln (1) = 1/a
Thus, we have shown that the derivative of ln x is 1/x.
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