Problem 35 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

\(F(u)=u^{2}+2 \ln (u+2)\)

Step-by-Step Solution

Answer

The first derivative is \( F'(u) = 2u + \frac{2}{u + 2} \), and the second derivative is \( F''(u) = 2 - \frac{2}{(u+2)^2} \).

Step 1 - Understand the Function

The given function is a combination of a polynomial and a logarithmic function: \[ F(u) = u^2 + 2 \ln (u + 2) \]Here, \( u^2 \) is a polynomial term, and \( 2 \ln (u + 2) \) is the logarithmic term.

Step 2 - Find the First Derivative

To analyze the function, find its first derivative. Use the power rule for \( u^2 \) and the chain rule for \( 2 \ln (u + 2) \). The first derivative is: \[ F'(u) = \frac{d}{du} (u^2) + \frac{d}{du} (2 \ln(u + 2)) \]Which simplifies to: \[ F'(u) = 2u + \frac{2}{u + 2} \]

Step 3 - Find the Second Derivative

Differentiate \( F'(u) \) to obtain the second derivative. Use the power rule and the chain rule as needed. The second derivative is: \[ F''(u) = \frac{d}{du} (2u) + \frac{d}{du} \left( \frac{2}{u + 2} \right) \]This simplifies to: \[ F''(u) = 2 - \frac{2}{(u+2)^2} \]

Key Concepts

First Derivative

The first derivative of a function gives us the rate at which the function's value changes with respect to changes in its input. In simpler terms, it tells us the slope of the function at any given point. Consider our function:

\[F(u) = u^2 + 2 \,\ln\,(u + 2)\]
To find the first derivative, we apply the power rule to the polynomial term and the chain rule to the logarithmic term. First, differentiate the polynomial part: \[\frac{d}{du}(u^2) = 2u\] Then, we handle the logarithmic part: \[\frac{d}{du}(2 \ln(u + 2))\] Here, we use the chain rule, where we differentiate the outer function (the logarithm) and multiply by the derivative of the inner function (u + 2): \[\frac{d}{du}(2 \ln(u + 2)) = 2 \cdot \frac{1}{u + 2} \cdot 1 = \frac{2}{u + 2}\] Finally, summing these derivatives, we get: \[F'(u) = 2u + \frac{2}{u + 2}\]
This is the first derivative of our function.

Second Derivative

The second derivative provides information about the curvature or concavity of the function. It indicates whether the function is concave up or concave down at a particular point. Consider the first derivative: \[F'(u) = 2u + \frac{2}{u + 2}\] To find the second derivative, we differentiate the first derivative: \[F''(u) = \frac{d}{du}(2u) + \frac{d}{du}\left( \frac{2}{u + 2} \right)\]
The derivative of 2u is simple: \[\frac{d}{du}(2u) = 2\] For the logarithmic part, we again use the chain rule: \[\frac{d}{du}\left( \frac{2}{u + 2} \right) = 2 \cdot \frac{d}{du}(u + 2)^{-1}\] Using the power rule: \[ \frac{d}{du}(u + 2)^{-1} = -\frac{1}{(u + 2)^2}\] Putting it together: \[F''(u) = 2 - \frac{2}{(u + 2)^2}\] This second derivative helps in understanding the behavior of the function's graph.

Power Rule in Calculus

The power rule is a basic principle used for differentiation in calculus. It is applied to functions of the form: \[f(x) = x^n\] It states that if you need to differentiate \[x^n\], then: \[\frac{d}{dx}(x^n) = nx^{n-1}\]
In our case, for the term \[ u^2\], we apply the power rule: \[\frac{d}{du}(u^2) = 2u\] This is essential for finding the first and second derivatives of functions involving polynomial terms. Whenever we have a term where the variable is raised to a power, the power rule simplifies the differentiation process.

Chain Rule in Calculus

The chain rule is a crucial concept when dealing with composite functions. It allows us to differentiate functions nested within other functions. For a function \[ h(x) = f(g(x))\], nested as \[f(g(x))\], the chain rule states: \[\frac{d}{dx}h(x) = f'(g(x)) \cdot g'(x)\]
In our exercise, applying the chain rule was necessary for the logarithmic term: \[2 \ln(u + 2)\] Since \[u + 2\] is within the logarithm, we use the chain rule: \[\frac{d}{du}(2 \ln(u + 2)) = 2 \cdot \frac{1}{u + 2} \cdot \frac{d}{du}(u + 2) = \frac{2}{u + 2}\] This simplifies the process of differentiating composite functions significantly and is an indispensable tool in calculus.