sin(3*x) --------- 5 cos (3*x)
d / sin(3*x)\ --|---------| dx| 5 | \cos (3*x)/
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 .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
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:
The result of the chain rule is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
2 3*cos(3*x) 15*sin (3*x) ---------- + ------------ 5 6 cos (3*x) cos (3*x)
/ 2 \ | 30*sin (3*x)| 9*|14 + ------------|*sin(3*x) | 2 | \ cos (3*x) / ------------------------------ 5 cos (3*x)
/ / 2 \\ | 2 | 42*sin (3*x)|| | 5*sin (3*x)*|17 + ------------|| | 2 | 2 || | 75*sin (3*x) \ cos (3*x) /| 27*|14 + ------------ + -------------------------------| | 2 2 | \ cos (3*x) cos (3*x) / -------------------------------------------------------- 4 cos (3*x)