36th Derivative of the Sine Fuction
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Find the
derivative of
.
How do we interpret the symbols
? Is multiplication involved?
No, this group of symbols describes a function named “
f
” which depends on the variable
x
.
We frequently use the variable “
y
” for our dependent variable. What is the dependent variable here?
The function
is the dependent variable here.
How do we proceed?
We determine the first, second, third, and fourth derivatives. Then we can usually see the pattern needed for the
derivative.
What rules of differentiation do we use here?
The Sine, Cosine, and Chain Rules.
Determine
.
Why do we have that last factor?
It indicates the application of the Chain Rule.
Simplify this expression.
We get
Determine
.
Simplify this expression.
We get
Determine
.
Simplify this expression.
We get
Determine
.
Simplify this expression.
We get
Let’s analyze what is happening here. Do we get a power of 3 each time?
Yes, because of the Chain Rule.
What is the pattern on the algebraic sign?
In succession, we have +, – , – , +
Will that repeat?
Yes.
What is the pattern on the trig function?
For the first four derivatives, we have
Will both of these patterns repeat?
Yes, so there is an overall pattern of groups of four derivatives.
To get the A36th derivative, we can exploit this cycle of repetition. Is 36 a multiple of 4?
Yes.
Which of the first four derivatives will
resemble?
The 4th ,
.
What power of 3 will it involve?
The 36th power.
Determine
Let’s determine the 19th derivative of
sin(3x). What power of 3 will be involved?
The 19th power.
How do we decide on the algebraic sign?
We determine where this derivative is in the cycle of 4 derivatives.
What is the nearest multiple of four, compared to 19?
20 is the nearest.
So 19 is one before 20. Which of our first four derivatives is one before the multiple of 4?
The third derivative is one before the fourth. That enables us to determine both the algebraic sign and the trig function in the 19th derivative from those of the third derivative.
Use these results to determine the 19th derivative of .
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know:
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General Contents
Detailed Contents
Index