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Calculus : Differentiate

Basic | Advanced | Help

The differentiate command will carry out ordinary or partial differentiation on virtually any expression.

By default, the differentiate command treats all variables in the expression, apart from those you are differentiating by, as constants. You can differentiate with respect to a variable n times by including a comma and the number n after the variable in the variables text area. For example, to differentiate an expression with respect to x three times, you would enter x,3 in the variables text area.

The advanced differentiate command enables you to differentiate with respect to any number of variables, any number of times. Simply enter each variable on a separate line. As before, multiple derivatives are indicated by following the variable with a comma and a number. The advanced differentiate command also allows you to specify any dependencies of functions which appear in your expression. The differentiate command handles arbitrary functional dependencies properly using the chain rule.

Examples

Here are some examples illustrating the types of expressions you can use the differentiate command on and the results which QuickMath will return.

Basic differentiate command

Expression Variable(s) Result
x^2 x 2 x
x^3 x
   2
3 x
5 x^3 - 7 x^2 + 2 x - 1 x
               2
2 - 14 x + 15 x
5 x^3 - 7 x^2 + 2 x - 1 x,2 -14 + 30 x
5 x^3 - 7 x^2 + 2 x - 1 x,3 30
sin(t) t cos(t)
sin(t) cos(t) t
      2         2
cos(t)  - sin(t)
ln(x)y + 3x^2y^3 x
y        3
- + 6 x y
x 

Advanced differentiate command

Expression Variable(s) Function(s) Result
6 x^3 y^2 + 3 x y^4

- 9 x^2 sin(y)
x
y
 
    2  2      4
18 x  y  + 3 y  - 18 x sin(y)
6 x^3 y^2 + 3 x y^4

- 9 x^2 sin(y)
x,2
y,3
  18 cos(y)
u v x u(x)
v(x)
v(x) u'(x) + u(x) v'(x)
u / v x u(x)
v(x)
v(x) u'(x) - u(x) v'(x)
-----------------------
             2
         v(x)
z t z(x,y)
x(t)
y(t)
       (0,1) 
y'(t) z     (x(t), y(t)) 

         (1,0)
+ x'(t) z     (x(t), y(t))

Options (advanced page only)

Functions

Values : checked or unchecked + empty string or list of functions with their dependencies
Default : unchecked + empty string

The functions option allows you to specify the dependencies of any arbitrary functions which appear in the expression being differentiated.

For instance, if the expression contains a function f which depends on x, then you would enter f(x) in the functions text area. The function itself should only be referred to as f within the expression, not f(x), as QuickMath has no way of knowing whether f(x) in an expression represents a function or the product f*x.

Functions can also depend on other functions. For instance, suppose that f depends on both x and y, whilst x and y themselves depend on t. Then you would enter

f(x,y)
x(t)
y(t)

in the functions text area, but refer to the functions simply as f, x and y within the expression itself.

If you use arbitrary functions within your expression, there may be derivatives in the answer returned by QuickMath. For instance, the term

f'(x)

in an answer indicates the first (ordinary) derivative of the function f with respect to x, whilst

 (1,0)
z     (x, y)
indicates the first (partial) derivative of z(x,y) with respect to x.