Question

In Exercises 37 and 38 , find the slope of the tangent line to the graph of the logarithmic function at the point (1,0) . \(y=\ln x^{3 / 2}\)

Short answer

The slope of the tangent line to the curve \(y = \ln x^{3 / 2}\) at the point (1,0) is \(3/2\).

Step-by-step solution

Differentiating the function

To find the slope of the tangent line to the graph of the function at a certain point, we first need to find the derivative of the function. The derivative of \(y=\ln x^{3/2}\) can be found using the chain rule of differentiation. We get \(y' = \frac{1}{x^{3/2}} \cdot \frac{3}{2} \cdot \sqrt{x}\). After simplification, we get \(y' = \frac{3}{2\sqrt{x}}\).

Evaluate Derivative at the Given Point

Now that we have the derivative, we can find its value at the point (1,0) to get the slope of the tangent line at that point. Hence, \(y'(1) = \frac{3}{2 \cdot 1} = \frac{3}{2}\).

Conclusion

Therefore, the slope of the tangent line to the curve \(y = \ln x^{3 / 2}\) at the point (1,0) is \(3/2\).

Key concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. They can be recognized by their characteristic form, \( y = \log_b(x) \), where \( x \) is the argument, and \( b \) is the base of the logarithm. In calculus, the natural logarithm \( \ln(x) \)—which has a base \( e \) (Euler's number, roughly equal to 2.71828)—is particularly important due to its properties and its appearance in various equations and applications.

When working with logarithmic functions, several key properties are useful to remember. For example, \( \ln(x^n) = n \cdot \ln(x) \) and \( \ln(e^x) = x \). These properties are instrumental when differentiating logarithmic expressions, as they allow you to manipulate the function into a form that makes it much easier to differentiate. It's also worth noting that the derivative of \( \ln(x) \) with respect to \( x \) is \( \frac{1}{x} \), a formula that is essential for exercises involving the slope of the tangent line on logarithmic functions.

Chain Rule of Differentiation

The chain rule is a fundamental tool in calculus used to calculate the derivative of a composition of functions.

The general form of the chain rule states that for two functions \( u(x) \) and \( v(x) \) where \( y = u(v(x)) \), the derivative of \( y \) with respect to \( x \) is \( y' = u'(v(x)) \cdot v'(x) \). This crucial technique allows for the differentiation of complex expressions and is particularly relevant when the function involves terms raised to a power, products, or quotients, and of course, logarithms. Calculus students should practice applying the chain rule in various situations to become proficient with it, as seen in the exercise involving \( y = \ln(x^{3/2}) \) where it's necessary to understand both the outer function (the natural logarithm) and the inner function (the power of \( x \)).

Understanding how to break down a function into its components and use the chain rule will enable you to differentiate functions that may at first glance seem complicated. Through this process, you can find the rate at which a function changes, which directly relates to the concept of slopes and tangents in the context of graphical analysis.

Derivative Evaluation

Evaluating the derivative of a function at a specific point is just as crucial as finding the derivative itself. This evaluation gives us the slope of the tangent line to the graph of the function at that exact point. The slope is a measure of how steep the line is, which corresponds to how rapidly the function value changes at that point.

In practice, once you have the derivative, substitute the x-coordinate of the point in question into the derivative to find the slope. For instance, in the given exercise, evaluating the derivative \( y' \) at the point \( (1,0) \) involves substituting \( x = 1 \) into \( y' \) to obtain the slope \( \frac{3}{2} \). This process is simple yet vital, as it allows for the determination of the instantaneous rate of change, leading to applications in motion, optimization, and other areas within and beyond mathematics. Remember, the precision in evaluating the derivative is key when you wish to understand the behavior of the function at any specific point on its graph.