Problem 4 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find the slope of the tangent line to the exponential function at the point \((0,1)\).

Step-by-Step Solution

Answer

The slope of the tangent line at the point (1,0) is 1/2.

Step 1: Finding the Derivative

The derivative of \( y = \ln x^{1 / 2} \) can be found using chain rule and derivative properties of logarithmic and power functions. So, the derivative will be: \( y' = \frac{1}{2x} * \frac{1}{x^{1/ 2}} \). When simplified, the derivative will become: \( y' = \frac{1}{2x^{3/2}} \).

Step 2: Finding the Slope

Substitute \( x = 1 \) into the derivative \( y' \). The slope will therefore be: \( y'(1) = \frac{1}{2(1)^{3/2}} \). After Simplifying this we get our slope as 1/2 .

Key Concepts

Derivative of Logarithmic Functions

Understanding the derivative of logarithmic functions is essential for calculus students, especially when it comes to graphing and analyzing the behavior of these functions. The general derivative of a logarithmic function, such as the natural logarithm, is given by the formula \( \frac{d}{dx}(\ln g(x)) = \frac{1}{g(x)} \cdot g'(x) \), where \( g(x) \) is a differentiable function of \( x \).
When a logarithm involves a power, such as \( \ln x^{1/2} \), it's helpful to remember that the logarithm of a power can be rewritten using the logarithm rule: \( \ln x^{a} = a \ln x \).This simplifies differentiating because you can then take the derivative of \( a \ln x \) directly. The derivative will involve a multiplication by the inside function’s derivative, which is referred to as the chain rule—a concept we'll touch upon in the next section.
In our exercise, the derivative of \( \ln x^{1/2} \) simplifies to \( \frac{1}{2x^{3/2}} \), which is the rate of change of the function with respect to x. Being familiar with these rules makes finding the derivatives of more complex logarithmic functions a more straightforward process.

Chain Rule in Calculus

The chain rule is a cornerstone of calculus, unlocking the ability to find derivatives of composite functions. Essentially, when you have a composite function \( f(g(x)) \), the chain rule states that the derivative is the product of the derivative of the outer function evaluated at the inner function \( f'(g(x)) \) and the derivative of the inner function \( g'(x) \).

Applying the Chain Rule

When we apply the chain rule to our problem where the function \( y \) is \( \ln x^{1/2} \), we recognize \( x^{1/2} \) as our inner function \( g(x) \). Differentiating \( \ln g(x) \) gives us \( \frac{1}{g(x)} \) as the derivative of the outer function, and \( \frac{d}{dx}x^{1/2} = \frac{1}{2x^{1/2}} \) as the derivative of our inner function. Multiplying these together yields \( \frac{1}{2x^{3/2}} \).
Understanding how to break down a function into its composite parts and apply the chain rule is crucial. It enables students to navigate through more complicated derivatives that would otherwise be very challenging to manage.

Finding Slope at a Point

In calculus, finding the slope of the tangent line at a specific point on a graph involves evaluating the derivative of the function at that point. The slope of the tangent line is essentially the instantaneous rate of change of the function at the given point, which is what the derivative represents at a fundamental level.

Calculating the Slope Using the Derivative

In our example, we first determined the derivative of the function \( y = \ln x^{1/2} \) as \( y' = \frac{1}{2x^{3/2}} \). To find the slope at the point \( (1,0) \), we simply substitute \( x = 1 \) into our derivative to get \( y'(1) = \frac{1}{2(1)^{3/2}} = \frac{1}{2} \). This calculated value, \( \frac{1}{2} \), is the slope of the tangent to our function at the point \( x = 1 \).
This method can be applied universally—whether the function is polynomial, trigonometric, logarithmic, or any other differentiable function. Grasping how to calculate the slope at a point is a powerful tool in a student's mathematical arsenal, paving the way for deeper understanding of function behavior and graphical interpretations.