Use the Calculus > Differentiate command or press Ctrl+Shift+D to find the partial derivative of an expression.  This command allows you to select the differentiation variable and the order of the derivative.


Alternatively, the partial derivative of the expression u with respect to the variable x can be entered by typing in an expression of an expression of the form DIF(u, x). Note that the differentiation must be with respect to a variable rather than an expression.  For example,

DIF(SIN(a·x^2),x)

displays as

d         2 
—— SIN(a·x )
dx          

and simplifies to

             2 
2·a·x·COS(a·x )


The nth order partial derivative of the expression u with respect to the variable x can be entered by typing in an expression of the form DIF(u, x, n).  For example, the second-order partial derivative

DIF(LN(COS(x)),x,2)

simplifies to

      1    
- —————————
         2 
   COS(x)  


You can issue successive Calculus > Differentiate commands or enter nested calls on DIF to obtain mixed partial derivatives.  For example, the mixed second-order partial derivative

DIF(DIF((a·x + b·y)^3,x),y)

simplifies to

6·a·b·(a·x + b·y)


If an expression does not explicitly contain the differentiation variable, it is considered independent of the variable and the derivative simplifies to 0.  To take the derivative of an arbitrary function, say F, first use the Author > Function Definition command1ER8JVI to declare F an arbitrary function.  For example, if F is declared an arbitrary function using the assignment

F(x) :=

then

DIF(F(x)^3,x)

simplifies to

            2
3·F'(x)·F(x) 


As shown above, the prime notation can be used to indicate differentiation of named functions.  If F is the name of a user-defined function of one variable, the nth derivative of F(x) with respect to u can be entered by typing an expression of the form F’...’(u), where ‘...’ is n primes.  For example, if F(x) is defined using the assignment

F(x) := SIN(a·x^2)

then F'(x) simplifies to

             2 
2·a·x·COS(a·x )

and F'(LN(y)) simplifies to

                     2 
2·a·LN(y)·COS(a·LN(y) )



Derive uses the chain rule to differentiate compositions of functions.  For example, F and G are declared arbitrary functions, then

DIF(F(G(x)),x)

simplifies to

G'(x)·F'(G(x))


Derive can compute the derivative of all its pre-defined mathematical functions and operators with the exception of the factorial operator ! (see Probability FunctionsProbability_Functions).  The derivative of ! can be defined in terms of the psi and polygamma functions defined in the ProbabilityFunctions.mth0FMTUG utility file.


Functions for numerical differentiation are defined in the NumericalApproximation.mthY5C_1X utility file.  Many applications of differentiation, including functions for implicit differentiation, curvature, tangents, and osculating circles, are defined in the DifferentiationApplications.mthAUKXVG utility file.


Other Calculus commandsCalculus_commands 

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