5  Lab 5: diff of log(x) and exp(x)

Objective

The objective of this lab is to deepen understanding of the differentiation of logarithmic and exponential functions, using both manual calculations and symbolic computation with Maxima. Students will:

  1. Derive the Natural Logarithm Function:
    • Calculate the derivative of the natural logarithm function by hand.
    • Use Maxima to verify the manually calculated derivative of the natural logarithm.
    • Understand the steps involved in deriving the natural logarithm function, including the application of exponential functions and implicit differentiation.
  2. Derive Exponential Functions:
    • Calculate the derivative of an exponential function with any base by hand.
    • Use Maxima to verify the manually calculated derivative of the exponential function.
    • Understand the steps involved in deriving exponential functions, including the application of logarithmic functions and implicit differentiation.
  3. Simplify Derivatives Using Logarithmic Properties:
    • Simplify the calculation of the derivative of rational functions using logarithmic properties.
    • Apply logarithmic properties to transform and simplify the differentiation process.
    • Use Maxima to perform these transformations and differentiate the simplified expressions.
  4. Compare Manual and Maxima Calculations:
    • Solve problems involving the differentiation of logarithmic and exponential functions by hand.
    • Use Maxima to solve the same problems and compare results.
    • Develop a step-by-step list of instructions for solving these problems manually to reinforce understanding of the techniques involved.
  5. Practice and Apply Skills:
    • Find three problems from the textbook that involve derivatives of logarithmic or exponential functions.
    • Solve these problems using Maxima.
    • Provide a detailed list of instructions for solving each problem by hand, demonstrating a clear understanding of the differentiation techniques and their applications.

By the end of this lab, students will have a comprehensive understanding of the differentiation of logarithmic and exponential functions, proficiency in using Maxima for symbolic computation, and enhanced problem-solving skills in calculus.

5.1 Let’s calculate the derivative of natural log by hand andcompare with maxima

5.1.1 a). Smart thing to do:

(%i1) eq: y = log(x);

\[\mathtt{(\textit{%o}_{1})}\quad y=\log x\]

(%i2) eq1: exp(lhs(eq)) = exp(rhs(eq));

\[\mathtt{(\textit{%o}_{2})}\quad e^{y}=x\]

5.1.2 b) Wappaa!!

(%i3) depends(y,x);

\[\mathtt{(\textit{%o}_{3})}\quad \left[ y\left(x\right) \right] \]

(%i4) eq2: diff(eq1,x);

\[\mathtt{(\textit{%o}_{4})}\quad e^{y}\,\left(\frac{d}{d\,x}\,y\right)=1\]

5.1.3 c) Let’s solve for dydx

(%i5) eq3: solve(eq2,'diff(y,x));

\[\mathtt{(\textit{%o}_{5})}\quad \left[ \frac{d}{d\,x}\,y=e^ {- y } \right] \]

5.1.4 d) Let’s express the derivative only as function of x

from eq1 we know who is \(e^y \cdots\)

(%i6) eq4: lhs(first(eq3))=1/x;

\[\mathtt{(\textit{%o}_{6})}\quad \frac{d}{d\,x}\,y=\frac{1}{x}\]

5.1.5 e) Maxima do know how to calculate the derivative of natural log function

(%i7) diff(log(x),x,1);

\[\mathtt{(\textit{%o}_{7})}\quad \frac{1}{x}\]

5.2 Let’s calculate the derivative of any exponential function by hand and compare with maxima

5.2.1 a). Smart thing to do:

(%i8) eq: y = b^(x);

\[\mathtt{(\textit{%o}_{8})}\quad y=b^{x}\]

(%i9) eq1: log(lhs(eq)) = log(rhs(eq) );

\[\mathtt{(\textit{%o}_{9})}\quad \log y=\log b\,x\]

5.2.2 b) Wappaa!!

(%i10) depends(y,x);

\[\mathtt{(\textit{%o}_{10})}\quad \left[ y\left(x\right) \right] \]

(%i11) eq2: diff(eq1,x);

\[\mathtt{(\textit{%o}_{11})}\quad \frac{\frac{d}{d\,x}\,y}{y}=\log b\]

5.2.3 c) Let’s solve for dydx

(%i12) eq3: solve(eq2,'diff(y,x));

\[\mathtt{(\textit{%o}_{12})}\quad \left[ \frac{d}{d\,x}\,y=\log b\,y \right] \]

5.2.4 d) Let’s express the derivative only as function of x

from eq we know who is \(y\cdots\)

(%i13) eq4: lhs(first(eq3))=log(b)*b^x;

\[\mathtt{(\textit{%o}_{13})}\quad \frac{d}{d\,x}\,y=b^{x}\,\log b\]

5.2.5 e) Maxima do know how to calculate the derivative of any exponential function of base \(b\).

(%i14) diff(b^x,x,1);

\[\mathtt{(\textit{%o}_{14})}\quad b^{x}\,\log b\]

5.3 Let’s use log properties to make simpler the calculation of derivative of rational functions.

5.3.1 a) Let’s define our rational function

(%i15) eq: y =(x^2+1)*%e^x/(x^2+4);

\[\mathtt{(\textit{%o}_{15})}\quad y=\frac{\left(x^2+1\right)\,e^{x}}{x^2+4}\]

5.3.2 b) Let’s calculate the log to the whole equation:

(%i16) eq2: log(eq);

\[\mathtt{(\textit{%o}_{16})}\quad \log y=\log \left(\frac{\left(x^2+1\right)\,e^{x}}{x^2+4}\right)\]

5.3.3 c) let’s apply log properties

(%i17) eq3: eq2,logexpand=super;

\[\mathtt{(\textit{%o}_{17})}\quad \log y=-\log \left(x^2+4\right)+\log \left(x^2+1\right)+x\]

5.3.4 d) Wappaa!!

(%i18) eq4: diff(eq3,x);

\[\mathtt{(\textit{%o}_{18})}\quad \frac{\frac{d}{d\,x}\,y}{y}=-\left(\frac{2\,x}{x^2+4}\right)+\frac{2\,x}{x^2+1}+1\]

5.3.5 e) Let’s solve for dydx

(%i19) eq5: solve(eq4,'diff(y,x));

\[\mathtt{(\textit{%o}_{19})}\quad \left[ \frac{d}{d\,x}\,y=\frac{\left(x^4+5\,x^2+6\,x+4\right)\,y}{x^4+5\,x^2+4} \right] \]

5.4 Find three problems from the book where derivatives of logarithm or exponential functions are need it.

5.4.1 a) Solve it by using maxima

5.4.2 b) Give a list of instructions for each problems, of how you will solve it by hand.

5.5 For each of the previous sections:

  • take pictures of your handwritten work, submit them to the dropbox.
  • Link each picture to the corresponding section of your wxmaxima document1 (Cell -> Insert Image).
  • be sure your code run correctly by pressing Shift+Enter in each code chunk.
  • be sure the whole document display correctly, by running Ctrl+R.
  • Submit on the corresponding Dropbox folder, with the corresponding name (FINAL_Lab5_Calc1_YOURLASTNAME_mmddyy.wxmx.)

  1. pictures need to be in the same folder of your .wxmx file↩︎