d/dx[lnx]
To find the derivative of the natural logarithm function, ln(x), with respect to x, we can use the chain rule
To find the derivative of the natural logarithm function, ln(x), with respect to x, we can use the chain rule. The chain rule states that if we have a composition of functions, f(g(x)), then the derivative of that composition is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
Considering ln(x) as f(g(x)), where f(u) = ln(u) and g(x) = x, we can find the derivative.
Let’s start by finding the derivative of the inner function g(x) = x. In this case, the derivative of g(x) with respect to x is simply 1. So, dg(x)/dx = 1.
Next, we need to find the derivative of the outer function f(u) = ln(u). The derivative of ln(u) with respect to u is 1/u. In our case, since u is x, the derivative is 1/x.
Finally, we multiply the derivative of the outer function by the derivative of the inner function:
d/dx[ln(x)] = d/du[ln(u)] * d/dx[x]
Since du/dx = 1 (as we found earlier), we can simplify:
d/dx[ln(x)] = 1/x * 1
Therefore, the derivative of ln(x) with respect to x is 1/x.
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