Understanding the Derivative of ln(x) Using Differentiation Rules

Derivative of ln(x)

To find the derivative of ln(x), we can use the rules of differentiation

To find the derivative of ln(x), we can use the rules of differentiation. Let us denote ln(x) as y.

First, we need to express ln(x) in terms of e. Recall that ln(x) is the natural logarithm of x, which means it represents the power to which e (Euler’s number, approximately 2.71828) must be raised to obtain x. So, we can express ln(x) as:

y = ln(x) = e^y = x

Now, let’s take the derivative of both sides with respect to x using the chain rule:

dy/dx = d(e^y)/dx = d(x)/dx

Using the chain rule, the derivative of e^y with respect to x is e^y times the derivative of y with respect to x (dy/dx). The derivative of x with respect to x is simply 1.

Therefore, we have:

dy/dx = e^y * dy/dx = 1

Now, we need to solve for dy/dx. We can rewrite e^y as x:

dy/dx = e^y * dy/dx = 1
dy/dx = x * dy/dx = 1
dy/dx / dy/dx = 1 / x
1 = 1 / x

Thus, the derivative of ln(x) is:

dy/dx = 1 / x

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