How to Find the Derivative of ln(x) using the Chain Rule and Logarithmic Properties

Derivative of ln

To find the derivative of ln(x), where ln represents the natural logarithm, we can use a property of logarithmic functions and the chain rule

To find the derivative of ln(x), where ln represents the natural logarithm, we can use a property of logarithmic functions and the chain rule.

The natural logarithm is the logarithm base e, where e is a constant approximately equal to 2.71828.

We can rewrite ln(x) as ln(e^x), since e^x is the inverse function of ln(x) and represents the exponentiation of e to the power of x.

Now, let’s differentiate ln(e^x) using the chain rule.

Let u = e^x
So, ln(e^x) can be written as ln(u)

The chain rule states that if we have y = f(g(x)), the derivative of y with respect to x is dy/dx = f'(g(x)) * g'(x). In this case, f(u) = ln(u) and u = e^x.

First, let’s find the derivative of f(u) = ln(u) with respect to u, denoted as f'(u).

The derivative of ln(u) with respect to u is 1/u.

Now, let’s find the derivative of u = e^x with respect to x, denoted as u’.

The derivative of e^x with respect to x is e^x, since e^x is its own derivative.

Now, applying the chain rule, we have:

dy/dx = f'(u) * u’
= (1/u) * e^x

Replacing u with e^x, we get:

dy/dx = (1/e^x) * e^x
= 1

Therefore, the derivative of ln(x) with respect to x is 1.

In summary, the derivative of ln(x) is 1.

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