If y=cos^2x-sin^2x, then y'=
a) -1
b) 0
c) -2(cosx+sinx)
d) 2(cosx+sinx)
e) -4(cosx)(sinx)
I thought the answer was C but the answer key says it is E. Please help. Thanks in advance.
To find the derivative of y = cos^2(x) - sin^2(x), you can use the chain rule and the power rule for differentiation.
Let's break down the equation step by step:
1. Rewrite y = cos^2(x) - sin^2(x) as y = (cos(x))^2 - (sin(x))^2.
2. Apply the power rule for differentiation, which states that if y = (f(x))^n, then y' = n(f(x))^(n-1) * f'(x).
- For the first term, (cos(x))^2, the power is 2, so its derivative is y1' = 2(cos(x))^(2-1) * -sin(x).
- For the second term, (sin(x))^2, the power is 2, so its derivative is y2' = 2(sin(x))^(2-1) * cos(x).
3. Simplify the derivatives:
- y1' = 2cos(x) * -sin(x) = -2cos(x)sin(x).
- y2' = 2sin(x) * cos(x) = 2cos(x)sin(x).
4. Combine the derivatives:
- y' = y1' - y2' = -2cos(x)sin(x) - 2cos(x)sin(x) = -4cos(x)sin(x).
Therefore, the correct answer is e) -4(cos(x))(sin(x)).