Derivative of loga(x)
The derivative of the natural logarithm function, ln(x), is 1/x, where x is a positive real number
The derivative of the natural logarithm function, ln(x), is 1/x, where x is a positive real number. However, when dealing with logarithms in other bases, such as a, the derivative is slightly more complicated.
The derivative of loga(x), where a is a positive real number other than 1, can be found using the change of base formula:
loga(x) = ln(x) / ln(a)
We can differentiate both sides of this equation with respect to x using the quotient rule.
Let’s start by finding the derivative of ln(x):
d/dx(ln(x)) = 1/x
Now, let’s find the derivative of ln(a):
d/dx(ln(a)) = 0
Since ln(a) is simply a constant with respect to x (since a is a constant), its derivative is 0.
Now, to apply the quotient rule, we have:
d/dx(loga(x)) = [1/x * ln(a) – ln(x) * 0] / (ln(a))^2
Simplifying the equation further, we get:
d/dx(loga(x)) = ln(a) / (x * ln(a))^2
or
d/dx(loga(x)) = ln(a) / (x * ln(a))^2
So, the derivative of loga(x), where a is a positive real number other than 1, is ln(a) / (x * ln(a))^2.
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