log(sin(3*x - 1))
log(sin(3*x - 1))
Let .
The derivative of is .
Then, apply the chain rule. Multiply by :
Let .
The derivative of sine is cosine:
Then, apply the chain rule. Multiply by :
Differentiate term by term:
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 derivative of the constant is zero.
The result is:
The result of the chain rule is:
The result of the chain rule is:
Now simplify:
The answer is:
3*cos(3*x - 1) -------------- sin(3*x - 1)
/ 2 \ | cos (-1 + 3*x)| -9*|1 + --------------| | 2 | \ sin (-1 + 3*x)/
/ 2 \ | cos (-1 + 3*x)| 54*|1 + --------------|*cos(-1 + 3*x) | 2 | \ sin (-1 + 3*x)/ ------------------------------------- sin(-1 + 3*x)