Understanding the Derivative of the Logarithm Function | Simplified Explanation and Formula

dy/dx log base a of x

The derivative of the logarithm function with respect to the variable x is given by:

dy/dx (log base a of x)

Let’s denote the logarithm base a of x as y: y = logₐ(x)

To find the derivative, we’ll use the change of base formula for logarithms:

logₐ(x) = log(x) / log(a)

Now, let’s differentiate both sides with respect to x

The derivative of the logarithm function with respect to the variable x is given by:

dy/dx (log base a of x)

Let’s denote the logarithm base a of x as y: y = logₐ(x)

To find the derivative, we’ll use the change of base formula for logarithms:

logₐ(x) = log(x) / log(a)

Now, let’s differentiate both sides with respect to x. We can use the quotient rule to differentiate the right-hand side:

d/dx [log(x) / log(a)] = [log(a) * d/dx(log(x)) – log(x) * d/dx(log(a))] / [log(a)]²

The derivative of log(x) with respect to x is simply 1/x, and the derivative of log(a) with respect to a is 0 since it’s a constant. Plugging these values back into the equation, we get:

dy/dx = [1/x * log(a) – log(x) * 0] / [log(a)]²

Simplifying the equation, we find:

dy/dx = log(a) / (x * log(a)) = 1 / x

Hence, the derivative of log base a of x with respect to x is 1/x.

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