Apply the quotient rule, which is:
and .
To find :
Let .
The derivative of sine is cosine:
Then, apply the chain rule. Multiply by :
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 of the chain rule is:
To find :
Let .
The derivative of cosine is negative sine:
Then, apply the chain rule. Multiply by :
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 of the chain rule is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
2*cos(2*x) 5*sin(2*x)*sin(5*x) ---------- + ------------------- cos(5*x) 2 cos (5*x)
/ 2 \ | 2*sin (5*x)| 20*cos(2*x)*sin(5*x) -4*sin(2*x) + 25*|1 + -----------|*sin(2*x) + -------------------- | 2 | cos(5*x) \ cos (5*x) / ------------------------------------------------------------------ cos(5*x)
/ 2 \ | 6*sin (5*x)| 125*|5 + -----------|*sin(2*x)*sin(5*x) / 2 \ | 2 | | 2*sin (5*x)| 60*sin(2*x)*sin(5*x) \ cos (5*x) / -8*cos(2*x) + 150*|1 + -----------|*cos(2*x) - -------------------- + --------------------------------------- | 2 | cos(5*x) cos(5*x) \ cos (5*x) / ------------------------------------------------------------------------------------------------------------- cos(5*x)