Mathematica Tutor

Part 9: Differentiation

Before you start, make sure your variables are cleared by entering

Clear[g, x]

1. Define g to be the function given by
   
   g(x) = x²cos x.

   Check your work by evaluating g(pi). You should obtain -pi². If you have trouble, look at Part 7 again.
   
   

2. Now enter
   D[g[x],x]
   Then enter
   D[g[x],x,x]
   How would you calculate the third derivative?
   
   Try this alternative way to get the third derivative:
   Enter
   D[g[x],{x,3}]
   
   
3. If you want to calculate the derivative of an expression that you have not yet entered, just replace g(x) by the expression. For example, enter
   D[x^3-x^2+2,x]
   Now insert a literal constant in the expression: Enter
   D[x^3-a*x^2+2,x]
   Then change the final x to an a. That is, enter
   D[x^3-a*x^2+2,a]
   What is the role of the symbol after the comma in the differentiation expression?
   
   
4. Now suppose you want the function dg/dx, that is, g'(x). If necessary, restore the definition of g by entering
   g[x_] := (x^2)*Cos[x]
   Then enter
   g'[x]
   Then
   g''[x]
   Evaluate the second derivative of g at 2 by entering
   N[g''[2]]
   Check your understanding so far by using Mathematica to calculate the second derivative of tan(x⁶-3x+5) at 3/2. (The value is approximately -4521.)