I believe your last step contains and error. Shouldn't the first parentheses contain (-12 +24x) ?
The "2" exponent outside the first bracket could have been written "cot^2"
I don't see any furher simplifications
y=cot³(1-2x)²
y'=3[cot(1-2x)²]² (-csc²(1-2x)²)(2-4x)(-2)
y'=(-6+24x)[cot(1-2x)²]²(-csc²(1-2x)²)
How do you simplify further?
The "2" exponent outside the first bracket could have been written "cot^2"
I don't see any furher simplifications
First, let's rewrite the expression:
y' = (-6 + 24x)[cot(1-2x)²]²(-csc²(1-2x)²)
Now, let's simplify each part of the expression:
1. Simplifying [cot(1-2x)²]²:
- Remember that cot(x) = 1/tan(x), and cot²(x) = 1/tan²(x).
- So, [cot(1-2x)²]² = [1/tan(1-2x)²]²
- Applying the rule (1/x)² = 1/x², we get:
[1/tan(1-2x)²]² = 1/[tan(1-2x)²] = 1/(tan²(1-2x))
2. Simplifying (-csc²(1-2x)²):
- Since csc(x) = 1/sin(x) and csc²(x) = 1/sin²(x), we have:
(-csc²(1-2x)²) = -1/sin²(1-2x)².
3. Putting it all together:
- Now we can substitute the simplified versions back into the original expression:
y' = (-6 + 24x) * (1/[tan²(1-2x)]) * (-1/sin²(1-2x)²)
The expression y' = (-6 + 24x) * (1/[tan²(1-2x)]) * (-1/sin²(1-2x)²) is the simplified form of the derivative.