Download and open the linked spreadsheet.  It contains three columns, (C - E), labeled x, f(x) = a*x², and f(x) = cos(bx).  The constants a and b can be changed in cells B1 and B2 respectively.
Let us find the derivative of f(x) = a*x², by approximating df(x)/dx ~ Δf(x)/Δx.

• Into cell F2 enter =(D3-D2)/(C3-C2).  (D3-D2) is the change in f(x) and (C3-C2) is the change in x from row 2 to row 3.  Copy the formula down to row 280.  Plot f(x) and Δf(x)/Δx versus x.  Describe your plots.

Now let us evaluate Δf(x)/Δx analytically.  Δf(x) = a(x₂² - x₁²) = a(x₂ + x₁)(x₂ - x₁).
Δx = x₂ - x₁.  Therefore Δf(x)/Δx = a(x₂ + x₁).
As the distance x₂ + x₁ becomes smaller and smaller the difference between between x₂ + x₁ becomes smaller and smaller, and when the points are infinitesimally close, x₂ becomes equal to x₁.  Then x₂ + x₁ = 2x₁ and Δf(x)/Δx = 2ax₁ when evaluated at x₁.

• Add a column containing 2ax to the spreadsheet.  Into cell G2 enter =2*$B$1*C2.  Copy the formula down to row 280.  Compare column F and column G.  Why is there a small difference in entries of the same rows?  Add a plot of 2ax versus x to your graph.
• Let us find the derivative of f(x) = cos(bx), by approximating df(x)/dx ~Δf(x)/Δx.
  • Into cell H2 enter =(E3-E2)/(C3-C2).  (E3-E2) is the change in f(x) and (C3-C2) is the change in x from row 2 to row 3.  Copy the formula down to row 280.  Plot f(x) and Δf(x)/Δx versus x.  Describe your plots.

  The analytically evaluated derivative of  Δf(x)/Δx is -b*sin(bx).

  • Add a column containing -b*sin(bx) to the spreadsheet.  Into cell I2 enter =-$B$2*SIN($B$2*C2).  Copy the formula down to row 280.  Compare column H and column I.  Add a plot of -b*sin(bx) versus x to your graph.