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Here's a little quiz to warm up. (Go to the tutorial on derivatives of powers if you want to review derivatives of powers of x.)
wrong, because the correct answer is  x^{4}  
wrong, because the correct answer is  3x^{2} 

wrong, because the correct answer is  3x^{2}  =  0  
wrong, because the correct answer is  ln x^{3}  
The above example suggests the following:
The derivative of a product is not the product of the derivatives.
Product and Quotient Rules
Product Rule
Product Rule In Words:
Quotient Rule
Quotient Rule In Words: 
Q Where do these rule come from?
A You can find a proof of the product rule in Section 4.1 of Applied Calculus, or Section 11.1 Finite Mathematics and Applied Calculus. for a proof of the quotient rule, press here.
Using the Product Rule
(There are examples similar to the following in Section 4.1 of Applied Calculus, or Section 11.1 Finite Mathematics and Applied Calculus. )
Let us find the derivative of
f(x) = (4x^{3}x^{4}) (11x    x ). 
Before we start, first recognize that f(x) is a product of two factors:
(4x^{3}x^{4}) 
and
(11x    x ). 
Therefore, the product rule applies. Before using the rule, let us first rewrite the function in exponent form:
Now we can apply the product rule. The rule says that:
dx 
[f(x)g(x)]  =  f'(x)g(x) + f(x)g'(x). 
Therefore,
dx 
(4x^{3}x^{4}) (11xx^{0.5} ) 
=  (12x^{2}4x^{3})(11xx^{0.5} ) + (4x^{3}x^{4})(110.5x^{0.5} ). 
(deriv. of first)(second left alone) + (first left alone) (deriv. of second)  
In some circumstances, it is helpful to simplfy the answer. Use your best judgement as the situation arises. Now here are some for you to try. You need not simply the answers for these.
Note Use proper graphing calculator format to input your answers (spaces are ignored). For example, input
Using the Quotient Rule
Next, let us calculate the derivative of
f(x)  =  x^{3} + x  . 
First, we recognize that f(x) is a quotient (one expression divided by another) and so we need to use the quotient rule:
dx 
g(x) 
=  [g(x)]^{2} 
dx 
x^{3} + x 
=  (x^{3} + x)^{2} 
That is the answer, although it is sometimes useful to simplify the numerator. In the question below, you will need to manipulate the answer a little in order to match the correct choice.
Q  The derivative of f(x)  =  x1  is ? 
Q This is all very well if what you're given is an obvious product or quotient. But we all know that instructors are fond of "inbetween" creatures, such as
(3x+1)  x^{2}+x  . 
Which rule do we use for that ??
A To deal with things like that  or any mathematical expression whatsoever, we use the following little secret desribed in Applied Calculus and Finite Mathematics and Applied Calculus, called the Calculation Thought Experiment:
Calculation Thought Experiment
The calculation thought experiment is a technique to determine whether to treat an algebraic expression as a product, quotient, sum, or difference. Given such an expression, consider the steps you would use in computing its value. If the last operation is multiplication, treat the expression as a product; if the last operation is division, treat the expression as a quotient, and so on. Using the Calculation Thought Experiment (CTE) to Differentiate a Function
Examples
1. (3x^{2}4)(2x+1) can be computed by first calculating the expressions in parentheses and then multiplying. Since the last step is multiplication, we can treat the expression as a product. 2. (2x1)/x can be computed by first calculating the numerator and denominator, and then dividing one by the other. Since the last step is division, we can treat the expression as a quotient. 3. x^{2} + (4x1)(x+2) can be computed by first calculating x^{2}, then calculating the product (4x1)(x+2), and finally adding the two answers. Thus, we can treat the expression as a sum. 4. (3x^{2}1)^{5} can be computed by first calculating the expression in parentheses, and then raising the answer to the fifth power. Thus, we can treat the expression as a power. 
Using the Calculation Thought Experiment (CTE)
Let us use the CTE to find the derivative of
(3x+1)  x^{2}+x  . 
To use this method, pretend you were calculating, one step at a time, the value of for, say, x = 5. (You don't need to actually do the calculation.) One way of doing the calculation would be to use the following procedure:
Since the last operation is multiplication, the CTE tells us that the given expression is a product and so we should use the product rule.
f(x) = (3x+1)  x^{2}+x 
. 
Thus,
f'(x)  =  dx 
(3x+1)  x^{2}+x 
+  dx 
x^{2}+x 
. . . . (I)  
(second left alone) + (first left alone)  (deriv. of second) 
Remember that the expressions "d/dx" are shorthand for "the derivative of ..." In other words, we haven't done the work yet; the line above is just telling us what we need to do. (If we wanted, we could take a coffee break and come back to it later to do the work.)
To finish the calculation, we must compute the magenta and bluecolored derivatives oneatatime and plug them in to the expression above:
The first (magenta) derivative is easy:
dx 
(3x+1) = 3 
dx 
x^{2}+x 
=  (x^{2}+x)^{2}  =  (x^{2}+x)^{2} 
Now substitute these derivatives into formula (I) to obtain the answer:
f'(x)  =  3  x^{2}+x 
+  (x^{2}+x)^{2} 
Whew !
Now you do one:
Q The Calculation Thought Experiment (CTE) tells us that
Q A valid first step in the calculation of the derivative of
4x^{2} +  3x 
is to write down:
8x  +  (3) 
dx 
(4x^{2})  +  dx 
3x 
8x  +  dx 
3x 
8x  +  9x^{2} 

For the next question, you need to enter an algebraic expression using proper graphing calculator format (spaces are ignored). For example, input
Q Finally, the derivative of  4x^{2} +  3x 
simplifies to 

Now try some of the exercises on Section 4.1 of Applied Calculus, or Section 11.1 of Finite Mathematics and Applied Calculus.