/ /1\\ -cos|x*sin|-|| \ \x//
-cos(x*sin(1/x))
The derivative of a constant times a function is the constant times the derivative of the function.
Let .
The derivative of cosine is negative sine:
Then, apply the chain rule. Multiply by :
Apply the product rule:
; to find :
Apply the power rule: goes to
; to find :
Let .
The derivative of sine is cosine:
Then, apply the chain rule. Multiply by :
Apply the power rule: goes to
The result of the chain rule is:
The result is:
The result of the chain rule is:
So, the result is:
Now simplify:
The answer is:
/ /1\ \ | cos|-| | | \x/ /1\| / /1\\ |- ------ + sin|-||*sin|x*sin|-|| \ x \x// \ \x//
2 / /1\ \ /1\ / /1\\ | cos|-| | sin|-|*sin|x*sin|-|| | \x/ /1\| / /1\\ \x/ \ \x// |- ------ + sin|-|| *cos|x*sin|-|| - -------------------- \ x \x// \ \x// 3 x
/ /1\ \ 3 | cos|-| | / /1\ \ /1\ / /1\\ /1\ / /1\\ | \x/ /1\| / /1\\ /1\ | cos|-| | cos|-|*sin|x*sin|-|| 3*sin|-|*sin|x*sin|-|| 3*|- ------ + sin|-||*cos|x*sin|-||*sin|-| | \x/ /1\| / /1\\ \x/ \ \x// \x/ \ \x// \ x \x// \ \x// \x/ - |- ------ + sin|-|| *sin|x*sin|-|| + -------------------- + ---------------------- - ------------------------------------------ \ x \x// \ \x// 5 4 3 x x x