91Ó°ÊÓ

Understanding how to differentiate logarithmic functions is crucial in calculus. These functions have the general form \( y = \ln(x^{{n}}) \) where \( n \) is a real number. To differentiate \( \ln(x^{{n}}) \) with respect to \( x \) we use the power rule in conjunction with the chain rule. A logarithmic function essentially measures the power to which a base, typically \( e \) (Euler's number), must be raised to obtain the number that is the argument of the logarithm.

In our example, the function \( y = \ln(x^{{1 / 2}}) \) can be differentiated by first rewriting it as \( y = \frac{1}{2}\ln(x) \) using logarithm properties. The derivative of \( \ln(x) \) with respect to \( x \) is \( 1/x \) and applying the constant multiple rule, we get \( dy/dx = \frac{1}{2}(1/x) \) or simply \( 1/(2x) \). This process demonstrates how combining various differentiation rules allows for the successful derivation of more complex functions.

Understanding and applying these concepts correctly is key to finding the rate of change of logarithmic functions at any given point, which is the essence of differentiation in calculus.

The chain rule is a fundamental tool in calculus for finding the derivative of a composition of functions. It states that if you have a function \( g \) which is composed of another function \( f \), represented as \( g(f(x)) \), then the derivative of \( g \) with respect to \( x \) is \( g'(f(x)) \times f'(x) \). This means you multiply the derivative of \( g \) with respect to \( f(x) \) by the derivative of \( f \) with respect to \( x \).

In the context of logarithmic functions, such as \( y = \ln(x^{1 / 2}) \), we identify \( g \) as the log function and \( f(x) \) as \( x^{1 / 2} \). Applying the chain rule, we determine the derivative of the outer function \( g \) with respect to \( f(x) \) and then multiply it by the derivative of \( f(x) \) with respect to \( x \). The chain rule simplifies the differentiation process, breaking down complex functions into manageable chunks that can be differentiated separately and then combined to find the overall derivative.

The point-slope form is an equation used to describe the line that has a specific slope and passes through a given point. It is expressed as \( y - y_{{1}} = m(x - x_{{1}}) \) where \( m \) is the slope of the line and \( (x_{{1}}, y_{{1}}) \) is the point through which the line passes. This formula is particularly useful when writing an equation of a tangent line to a function at a given point.

When we talk about a tangent line to a curve at a certain point, we refer to a line that just touches the curve at that point and has the same slope as the curve at that point. The slope of the tangent line at any point on a function is simply the value of the derivative of that function at that point. Once we have the slope and we have a point on the line (in our case, \( (1,0) \) on the curve \( y = \ln(x^{1 / 2}) \) ), we can plug these into the point-slope equation to get the equation of our tangent line: \( y = 1/2x - 1/2 \). This equation now represents the tangent line at the point \( (1,0) \) on our original function, providing a visual and mathematical representation of the behavior of the function at that specific point.