7.2: The Logarithmic Derivative
This activity is intended to supplement Calculus, Chapter 6, Lesson 3.
ID: 9092
Time required: 45 minutes
Activity Overview
Students will determine the derivative of the function \begin{align*}y = \ln (x)\end{align*}
Topic: Formal Differentiation
 Derive the Logarithmic Rule and the Generalized Logarithmic Rule for differentiating logarithmic functions.

Prove that \begin{align*}\ln (x) =\ln (a) \cdot \log_a(x)\end{align*}
ln(x)=ln(a)⋅loga(x) by graphing \begin{align*}f(x)=\log_a(x)\end{align*}f(x)=loga(x) and \begin{align*}g(x) = \ln (x)\end{align*}g(x)=ln(x) for some a and deduce the Generalized Rule for Logarithmic differentiation.  Apply the rules for differentiating exponential and logarithmic functions.
Teacher Preparation and Notes
 This investigation derives the definition of the logarithmic derivative. The students should be familiar with keystrokes for the limit command, the derivative command, entering the both the natural logarithmic and the general logarithmic functions, drawing a graph, and setting up and displaying a table.

Before starting this activity, students should go to the home screen and select \begin{align*}F6\end{align*}
F6 :Clean Up > 2:NewProb, then press \begin{align*}\div\end{align*}÷ . This will clear any stored variables, turn off any functions and plots, and clear the drawing and home screens.  This activity is designed to be studentcentered with the teacher acting as a facilitator while students work cooperatively.
Associated Materials
 Student Worksheet: Sum Rectangles http://www.ck12.org/flexr/chapter/9731, scroll down to the second activity.
Problem 1 – The Derivative for \begin{align*}y = \ln (x)\end{align*}y=ln(x)
In this problem, students are asked to use the limit command (\begin{align*}F3\end{align*}
The students should get \begin{align*}\frac{1}{x}\end{align*}
In this portion, the students are asked to use the derivative command (\begin{align*}F3\end{align*}
Problem 2 – The Derivative of \begin{align*}y = \log_a(x)\end{align*}y=loga(x)
Students should notice that both functions have a value of zero for \begin{align*}\ln (1)\end{align*}
When \begin{align*}\log_4(x)\end{align*}
Have the students notice that \begin{align*}\ln (4)\end{align*}
Students should graph the functions \begin{align*}y1 = \ln (x), y2 = \ln (2) \cdot \log_2(x), y3 = \ln (3) \cdot \log_3(x)\end{align*}
Students are asked to find the Generalized Logarithmic Rule for Differentiation. They should find \begin{align*}\frac{dy}{dx} = \frac{\log_a (e)}{x}\end{align*}
Problem 3 – Derivative of Exponential and Logarithmic Functions Using the Chain Rule
Students are asked to identify \begin{align*}u(x)\end{align*}
Recall: \begin{align*}y = a^u \rightarrow \frac{dy}{dx} = a^u \ \frac{du}{dx}\end{align*}

\begin{align*}y = \log_a(u) \rightarrow \frac{dy}{dx} = \frac{1}{(u \ln (a))} \cdot \frac{du}{dx}\end{align*}
y=loga(u)→dydx=1(uln(a))⋅dudx or \begin{align*}\frac{dy}{dx} = \frac{ \frac{du}{dx}}{u(\ln (a))}\end{align*}dydx=dudxu(ln(a))
\begin{align*}f(x) = 5^{(x^2)}, u(x) = x^2, a = 5 \end{align*}
 \begin{align*}f'(x) = 2 \cdot \ln (5) \cdot x \cdot 5^{(x^2)}\end{align*}
\begin{align*}g(x) = e^{(x^3 + 2)}, u(x) = x^3 + 2, a = e \end{align*}
 \begin{align*}g'(x) = 3 \cdot x^2 \cdot e^{(x^3 + 2)}\end{align*}
\begin{align*}h(x) = \log_3(x^4 + 7), u(x) = x^4 + 7, a = 3\end{align*}
 \begin{align*}h'(x) = \frac{4 \cdot x^3 \log_3(e)}{x^4 + 7}\end{align*}
\begin{align*}j(x) = \ln \left (\sqrt{x^6 + 2}\right ), u(x) = \sqrt{x^6 + 2}, a = e\end{align*}
 \begin{align*}j'(x) = \frac{3 \cdot x^5}{x^6 + 2}\end{align*}
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