Calculus can be Simple

At times, I found the OU's treatment of differentiation, a little contrived.  It was okay; and is probably the proper approach to learning the elements of calculus, preparing the way for Analysis at level 2 study.

However, I think much more emphasis should have also been given to teaching the mechanical, practical rules of differentiation and integration, in a much more utilitarian way.

I would like to have seen a separate booklet containing hundreds of progressive quick differentiation and integration problems, that could help towards mastery of the mechanical act of 'doing' the calculus.  I believe that there are budding scientists or engineers who don't care for the minutia of Analysis, but instead need to use it as a tool in their trade, who may be put off by a lack of 'practical' exercises.

Anyway, I thought I would share an extract of the book 'Calculus Made Easy, by Thompson, Gardner.  It just shows how simple calculus can be made.  What is striking about this extract, is that the OU tried to explain the exact same element of learning, in about 800 words, whereas Gardner does it in about 100 words.  Also, on first reading of the OU text, I didn't grasp the concepts and it took several hours of self study, to practice and attempt mastery of the technique.  I read the Gardner extract in approximately 4 minutes and was able to apply it immediately.  Here it is:

'Sometimes one is stumped by finding that the expression to be differentiated is too complicated to tackle directly. 

 Thus, the equation:   \(y = {({x^2} + {a^2})^{\frac{3}{2}}}\)  is awkward for a beginner.

Now the dodge to turn the difficulty is this:  

Write some symbol such as 'u' for the expression  \(({x^2} + {a^2})\)

then the equation becomes \(y = {u^{\frac{3}{2}}}\)

Which you can then easily manage; for  \(\frac{{dy}}{{du}} = \frac{3}{2}{u^{\frac{1}{2}}}\)

Then tackle the expression  \(u = ({x^2} + {a^2})\)

 and differentiate it with respect to x, thus,  \(\frac{{dy}}{{du}} = 2x\)

Then all that remains is plain sailing, for \(\frac{{dy}}{{dx}} = \frac{{dy}}{{du}} \times \frac{{du}}{{dx}}\)

     \(\frac{{dy}}{{dx}} = \frac{3}{2}{u^{\frac{1}{2}}} \times 2x\)

           \( = \frac{3}{2}{({x^2} + {a^2})^{\frac{1}{2}}} \times 2x\)     

           \( = 3x{({x^2} + {a^2})^{\frac{1}{2}}}\)     

And so the trick is done.'

Calculus Made Easy (1998)

Wonderfully simple.  Oh, how glad I am that I found this book.