Derivative of log base b of x
The derivative of the logarithm function, specifically the natural logarithm (base e), is a well-known result
The derivative of the logarithm function, specifically the natural logarithm (base e), is a well-known result. However, the derivative of the logarithm with respect to different bases, such as logarithm base b of x (log_b(x)), requires a slight modification in the formula.
To find the derivative of log_b(x), we need to use the change of base formula and the chain rule of differentiation.
The change of base formula states that log_b(x) can be written as log(x)/log(b), where log denotes the natural logarithm (base e). So, we can rewrite log_b(x) as log(x)/log(b).
Now, let’s find the derivative of log(x)/log(b) with respect to x by using the quotient rule and the chain rule.
First, we need to find the derivative of log(x) and log(b) separately:
1. Derivative of log(x):
The derivative of log(x) with respect to x is 1/x. This is one of the basic differentiation rules for logarithms.
2. Derivative of log(b):
Since log(b) is a constant term with respect to x, its derivative is zero.
Next, let’s apply the quotient rule:
Using the quotient rule: [d(u)/dx * v – u * d(v)/dx] / v^2
Where u = log(x) and v = log(b)
Applying the quotient rule:
[d(log(x))/dx * log(b) – log(x) * d(log(b))/dx] / [log(b)]^2
Simplifying further:
[(1/x) * log(b) – log(x) * 0] / [log(b)]^2
Simplifying the expression:
(1/x * log(b)) / [log(b)]^2
Finally, simplifying the expression:
1 / (x * log(b))
Therefore, the derivative of log_b(x) with respect to x is 1 / (x * log(b)).
This result gives us the rate of change of the logarithm with base b with respect to x.
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