Derivative of Logarithm in Alternative Bases: Formula and Calculation

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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