Calculate the derivative of the function y = log(x) base e.

The derivative of y = log(x) base e is 1/x.

To find the derivative of y = log(x) base e, we can use the formula for the derivative of a natural logarithm function:

dy/dx = 1/x

where x is the input variable. This formula holds true for any base of the logarithm, as long as the base is a positive constant.

To see why this formula works, we can use the definition of the natural logarithm:

ln(x) = y if and only if e^y = x

Taking the derivative of both sides with respect to x, we get:

d/dx(e^y) = d/dx(x)

Using the chain rule on the left-hand side, we get:

e^y * dy/dx = 1

Solving for dy/dx, we get:

dy/dx = 1/e^y

Substituting y = ln(x), we get:

dy/dx = 1/x

Therefore, the derivative of y = log(x) base e is 1/x.

For a foundational understanding of derivative concepts, including the natural logarithm, visit our Introduction to Derivatives page.

Additionally, to explore more about how derivatives apply to exponential and logarithmic functions, check out Differentiation of Exponential and Logarithmic Functions.

A-Level Maths Tutor Summary: In simpler terms, the rate at which the natural log function y = log(x) with base e changes as x changes is given by 1/x. This is because when we look at how the function grows, it increases at a pace that is inversely proportional to x. So, for the natural log of x, the quicker formula to find its derivative is simply 1/x. For more details on differentiating functions like these, consider our resource on Basic Differentiation Rules.