Calculus I
Notes on Combining Derivative Rules.
Many derivatives require the use of more than 1 derivative rules. To figure out how to take the derivative of more
complicated functions, you must consider order of operations and alternate forms.
Example #1: Differentiate f (x) 
sin(x) cos(x)
x4
►
Since we can’t simplify the expression, we need to focus on the order of operations. They are:
Level
1
2
3
Operation(s)
sine, cosine, and 4th power
product of sin(x) and cos(x)
quotient of numerator and denominator
When dealing with a combination of rules we have to deal with the operation from last to first. Thus, the order of the rules would be:
Level
1
2
3
Rule(s)
Quotient Rule
Product Rule
Misc. Basic Rules
Therefore,
4
d
d
 4
dx  sin(x) cos(x)   x  sin(x) cos(x)  dx  x 
f '(x) 
, Quotient Rule
2
 x4 
4
4
d
d
d
 dx sin(x)   cos(x)  sin(x )  dx  cos(x)   x  sin(x) cos(x )  dx  x 

, Product Rule
2
 x4 
 cos(x)   cos(x)  sin(x)   -sin( x )   x 4  sin( x) cos( x )   4x 3 

, Misc.Basic Rules
2
 x4 
cos 2 (x)  sin 2 (x)   x 4  4x 3 sin(x) cos(x) x 4 cos 2 (x)  x 4 sin 2 (x)  4x 3 sin(x) cos(x)


x8
x8
2
2
x cos (x)  x sin (x)  4sin(x) cos(x)

x5
□
Note that in the previous example, I could have started some of the basic rules and simplifying earlier, but I waited so to emphasize
the derivative rules. Later ones I’ll do in a more condensed form.
Example #2: Differentiate f (x)  tan 8x 5 
►
2
f (x)  tan 8x 5    tan 8x 5 
This one is a function within a function within a function. Thus, we have to use a chain rule within a chain rule.
1
f '(x)  12  tan 8x 5 

1 1
2


 d tan 8x 5    1  tan 8x 5 

 dx
 2
- 12


 sec 2 8x 5  40x 4   



  tan 8x 
tan  8x 
20x 4 sec2 8x 5
 
 
20x 4 sec2 8x 5
tan 8x 5
5
5
□
Seminole State:Rickman
Notes on Combining Derivative Rules.
Page #1 of 2
Example #3: Differentiate f (x) 
1
x cos(x)
6
►
If I did the derivative as is it would be a product rule within a quotient rule. Instead I’m going to use a different form of the function
to make the derivative easier.
1
f (x)  6
 x - 6 sec(x)
x cos(x)
f '(x)  -6x -7  sec(x)  x -6 sec(x) tan(x)   -6x -7 sec(x)  x -6 sec(x) tan(x)
□
Thus, you should always look for better forms for taking the derivative.
Example #4: Differentiate f (x)  3 x 5 cot 5  x 2 
►
3
f (x)  3 x 5 cot 5  x 2    x cot  x 2 
5
3
f '(x)  53  x cot  x 2   1 cot  x 2   x -csc  x 2   2x  


 53 3 x 2 cot 2  x 2  cot  x 2   2x 2 csc  x 2  
2
□
Example #5: Differentiate f (x) 
sin(3x)
sin(7x)
►
cos(3x) 3 sin(7x)  sin(3x) cos(7x) 7 
f '(x)  
sin 2 (7x)
3cos(3x)sin(7x)  7 sin(3x) cos(7x)

sin 2 (7x)
□
 x 5 
f (x)  sec 

3
 x 
Example #6: Differentiate
►
f '(x)  sec
  tan  
x 5
x3
x 5
x3
  12  x  5- 12 1 x3 


2
 x3 

 sec
  tan   
 sec
  tan   
x 5
x3
x 5
x3
x 5
x3
 - 52 sec
x 5
x3
x 3  6x 2  x  5
2x 6 x  5
-5x  30
2x 4 x  5
  tan    
x 5
x3
x 5
x3
x  5 3x 2 


  sec

  sec


  sec

  tan  
x 5
x3
x 5
x3
  tan   
x 5
x3
x 5
x3
  tan   
x 5
x3
x 5
x3

2

x 3  6x 3  30x 2
2x 6 x  5
-5 x  6  x  5
2x 4  x  5 
x3
x 5
 3x 2 x  5
x
6
  sec

 22
x 5
x 5



  tan   
x 5
x3
x 5
x3
-5x 3  30x 2
2x 6 x  5




x  6 x  5
x 4  x  5
□
Example #7: Differentiate f (x)   4x  7   3x  1
6
2
►
f '(x)  6  4x  7  [4]  3x  1   4x  7  2  3x  1[3]  24  4x  7  3x  1  6  4x  7  3x  1


5
 6  4x  7   3x  1  4  3x  1   4x  7  
5
 6  4x  7   3x  116x  11
□
5
Seminole State:Rickman
2
6
5
2
Notes on Combining Derivative Rules.
6
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