derivative of ln
The derivative of ln(x), where ln represents the natural logarithm, can be found using the rules of calculus
The derivative of ln(x), where ln represents the natural logarithm, can be found using the rules of calculus.
The natural logarithm ln(x) is the logarithm function with base e, where e is the mathematical constant approximately equal to 2.71828.
To find the derivative of ln(x), we can apply the chain rule. The chain rule states that if we have a composite function f(g(x)), then the derivative is given by f'(g(x)) * g'(x).
In this case, we can consider ln(x) as the composite function where g(x) = x and f(u) = ln(u).
Applying the chain rule, we have:
d/dx[ln(x)] = d/dx[ ln(g(x)) ] = f'(g(x)) * g'(x)
Now, let’s find the derivative of f(u) = ln(u). The derivative of ln(u) with respect to u is given by 1/u.
Therefore, f'(u) = 1/u.
Now, let’s find the derivative of g(x) = x. The derivative of x with respect to x is simply 1.
Therefore, g'(x) = 1.
Putting it all together, we have:
d/dx[ln(x)] = f'(g(x)) * g'(x) = (1/x) * 1 = 1/x
Hence, the derivative of ln(x) is 1/x.
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