log(sin(2*x + 5))
log(sin(2*x + 5))
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:
2*cos(2*x + 5) -------------- sin(2*x + 5)
/ 2 \ | cos (5 + 2*x)| -4*|1 + -------------| | 2 | \ sin (5 + 2*x)/
/ 2 \ | cos (5 + 2*x)| 16*|1 + -------------|*cos(5 + 2*x) | 2 | \ sin (5 + 2*x)/ ----------------------------------- sin(5 + 2*x)