sin(x) ------------- 2 2 + 3*cos (x)
Apply the quotient rule, which is:
and .
To find :
The derivative of sine is cosine:
To find :
Differentiate term by term:
The derivative of the constant is zero.
The derivative of a constant times a function is the constant times the derivative of the function.
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
The derivative of cosine is negative sine:
The result of the chain rule is:
So, the result is:
The result is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
2 cos(x) 6*sin (x)*cos(x) ------------- + ---------------- 2 2 2 + 3*cos (x) / 2 \ \2 + 3*cos (x)/
/ / 2 2 \ \ | | 2 2 12*cos (x)*sin (x)| | | 6*|cos (x) - sin (x) + ------------------| | | | 2 | 2 | | \ 2 + 3*cos (x) / 12*cos (x) | |-1 + ------------------------------------------ + -------------|*sin(x) | 2 2 | \ 2 + 3*cos (x) 2 + 3*cos (x)/ ------------------------------------------------------------------------ 2 2 + 3*cos (x)
/ / 2 2 2 2 \\ | / 2 2 \ 2 | 9*cos (x) 9*sin (x) 54*cos (x)*sin (x)|| | | 2 2 12*cos (x)*sin (x)| 24*sin (x)*|1 - ------------- + ------------- - ------------------|| | 18*|cos (x) - sin (x) + ------------------| | 2 2 2 || | 2 | 2 | | 2 + 3*cos (x) 2 + 3*cos (x) / 2 \ || | 18*sin (x) \ 2 + 3*cos (x) / \ \2 + 3*cos (x)/ /| |-1 - ------------- + ------------------------------------------- - -------------------------------------------------------------------|*cos(x) | 2 2 2 | \ 2 + 3*cos (x) 2 + 3*cos (x) 2 + 3*cos (x) / ----------------------------------------------------------------------------------------------------------------------------------------------- 2 2 + 3*cos (x)