How To Find The Derivative Of Ln(X) With Step-By-Step Guide?

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.

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