MATLAB can find the derivative of symbolic expressions. Higher derivatives can also be found easily. These derivatives can be evaluated for arbitary values of the independent variable.

1. First we declare x to be a symbolic variable and define g to be the function given by g(x) = x² cos x.   Enter:
   
   syms x
   g = x^2*cos(x)
   
   


2. Now enter:
   
   diff(g, x)
   
   Then enter:
   
   diff(g, x, 2)
   
   How would you calculate the third derivative?
   
   


3. If you want to calculate the derivative of an expression that you have not yet entered, just replace g by the expression. For example, enter:
   
   diff(x^3-x^2+2, x)
   
   Now insert a symbolic constant a into the expression.   Enter:
   
   syms a
   diff(x^3 - a*x^2 + 2, x)
   
   Then change the final x to an a. That is, enter:
   
   diff(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 to evaluate the derivative dg/dx at x = 2.1. Enter:
   
   gp=diff(g, x)
   
   Then enter:
   
   subs(gp, x, 2.1)
   
   Alternatively, you could enter:
   
   subs( diff(g, x), x , 2.1 )
   
   


5. Check your understanding so far by using MATLAB to calculate the second derivative of tan(x⁶ - 3x + 5) at x = 3/2.
   (The value is approximately  - 4521.)