cot(x) ------ 3 x - x
cot(x)/(x - x^3)
Apply the quotient rule, which is:
and .
To find :
There are multiple ways to do this derivative.
Rewrite the function to be differentiated:
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Rewrite the function to be differentiated:
Apply the quotient rule, which is:
and .
To find :
The derivative of sine is cosine:
To find :
The derivative of cosine is negative sine:
Now plug in to the quotient rule:
The result of the chain rule is:
Rewrite the function to be differentiated:
Apply the quotient rule, which is:
and .
To find :
The derivative of cosine is negative sine:
To find :
The derivative of sine is cosine:
Now plug in to the quotient rule:
To find :
Differentiate term by term:
Apply the power rule: goes to
The derivative of a constant times a function is the constant times the derivative of the function.
Apply the power rule: goes to
So, the result is:
The result is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
2 / 2\ -1 - cot (x) \-1 + 3*x /*cot(x) ------------ + ------------------ 3 2 x - x / 3\ \x - x /
/ / 2\ \ | | / 2\ | | | | \-1 + 3*x / | | | |3 - ------------|*cot(x) | | | 2 / 2\| / 2 \ / 2\| | / 2 \ \ x *\-1 + x // \1 + cot (x)/*\-1 + 3*x /| 2*|- \1 + cot (x)/*cot(x) + ------------------------- - -------------------------| | 2 / 2\ | \ -1 + x x*\-1 + x / / ---------------------------------------------------------------------------------- / 2\ x*\-1 + x /
/ / 3\ \ | / 2\ | / 2\ / 2\ | | | | / 2\ | | 6*\-1 + 3*x / \-1 + 3*x / | | | / 2 \ | \-1 + 3*x / | 3*|1 - ------------- + -------------|*cot(x) | | 3*\1 + cot (x)/*|3 - ------------| | 2 2| | | | 2 / 2\| | -1 + x 2 / 2\ | / 2 \ / 2\ | |/ 2 \ / 2 \ \ x *\-1 + x // \ x *\-1 + x / / 3*\1 + cot (x)/*\-1 + 3*x /*cot(x)| 2*|\1 + cot (x)/*\1 + 3*cot (x)/ - ---------------------------------- + -------------------------------------------- + ----------------------------------| | 2 / 2\ / 2\ | \ -1 + x x*\-1 + x / x*\-1 + x / / ---------------------------------------------------------------------------------------------------------------------------------------------------------- / 2\ x*\-1 + x /