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Lesson 5. Differentiation & Derivatives
The outcome of differentiation on a function is called the “derivative”. At a particular number a from the
domain of a function f, the derivative )(' af is a quantity, which is called the instantaneous rate of
change at a. On an interval at which a function fis defined, )(' xf is a function. In this lesson we start to
learn how to use Mathematica to find the derivatives and derivative functions of a function.
(1) First Derivative of a Function
There are two commonly used approaches to find the derivative of a function by Mathematica.
Example 1: Find the derivative of xxxf sin)( 2+= .
The 1st way: , Shift + Enter
The result:
[Note] The letter “D” stands for “Differentiating” . Inside [ ] the variable x must be input behind the
function and separated by a comma, which means the differentiation is taken with respect to x.
The 2nd way: _: ^ Shift +Enter
Shift +Enter
The result:
[Note] The symbol “ : ” stands for “Defining”. Defining a function of a variable x, it is a Mathematica
standard way to type in “_” ; Be careful that “Shift + Enter” can NOT be skipped when defining the
function )(xf . Pressing “Shift+Enter” is to validate this definition, otherwise it won’t show that result
the result of )(' xf .
There will be no difference between the two ways when finding the derivatives of a single-variable
function. But keep in mind that “ D[ ]” is the only way of finding a partial derivative of a multivariate
function, and ][' xf is basically a way of finding a derivative for a single-variable function. Note that
within the text of calculus I and II, we are not going to deal with any partial derivative of a multivariate
function.
