Derivative of a logarithmic function?
The derivative of a logarithmic function can be found using the rules of differentiation
The derivative of a logarithmic function can be found using the rules of differentiation. Let’s consider the logarithmic function with base b, denoted as f(x) = log_b(x).
To find the derivative of this logarithmic function, we can use the logarithmic differentiation rule which states that the derivative of log_b(x) with respect to x is given by:
f'(x) = (1 / (x * ln(b)))
Here, ln(b) represents the natural logarithm of the base b.
To understand how this derivative is derived, let’s go through the steps:
Step 1: Start by expressing the logarithmic function using the natural logarithm (ln):
f(x) = ln(x) / ln(b)
Step 2: Apply the logarithmic differentiation rule, which states that the derivative of ln(u) is equal to (1/u) multiplied by the derivative of u with respect to x. In our case, the derivative of ln(x) is (1/x):
f'(x) = [(1 / x) * ln(b) – ln(x) * (0)] / (ln(b))^2
= (ln(b) / x) / (ln(b))^2
= ln(b) / [x * (ln(b))^2]
= (1 / x) * (1 / ln(b))
= 1 / (x * ln(b))
Thus, the derivative of the logarithmic function f(x) = log_b(x) is f'(x) = (1 / (x * ln(b))).
It’s important to note that this rule for the derivative of a logarithmic function holds true for any base b. If the base is not specified, it is often assumed to be the natural logarithm (base e).
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