d/dx ln(x)
1/x
The derivative of the natural logarithm function ln(x) is given by:
d/dx ln(x) = 1/x
To see how this derivative is obtained, we need to use the definition of the derivative:
d/dx ln(x) = lim h -> 0 (ln(x + h) – ln(x)) / h
Using the logarithmic identity ln(a) – ln(b) = ln(a/b), we can simplify the numerator:
ln(x + h) – ln(x) = ln((x + h)/x)
We can then rewrite the limit as:
d/dx ln(x) = lim h -> 0 ln((x + h)/x) / h
Applying L’Hopital’s rule by taking the derivative of the numerator and denominator with respect to h, we get:
d/dx ln(x) = lim h -> 0 (1/x) / 1
Simplifying further, we obtain:
d/dx ln(x) = 1/x
Therefore, the derivative of ln(x) is 1/x.
More Answers:
[next_post_link]