Find derivative of:
g(theta) = cos^2(2theta-1)
g ' (Ø) = 2cos(2Ø -1) (2)
= ...
hmmm. power rule, then chain rule twice
yup, forgot about the exponent, Steve is right
g ' (x) = 2 cos(2Ø-1) (-sin(Ø-1)) ( 2)
= -4 sin(2Ø-1)cos(2Ø-1)
which they might have changed to
-2 sin(4Ø-2)
To find the derivative of g(theta) = cos^2(2theta-1), we can use the chain rule. The chain rule allows us to find the derivative of a composite function by differentiating the outer function and then the inner function.
Let's break down the steps to find the derivative:
Step 1: Identify the outer and inner functions.
In this case, the outer function is the square of the cosine function: cos^2(x), and the inner function is 2theta - 1.
Step 2: Differentiate the outer function.
The derivative of cos^2(x) is -sin(2x). This is a well-known derivative, so we don't need to go through the derivation steps.
Step 3: Differentiate the inner function.
The derivative of 2theta - 1 with respect to theta is 2.
Step 4: Apply the chain rule.
According to the chain rule, we need to multiply the derivative of the outer function with the derivative of the inner function. In this case, it will be:
- sin(2(2theta-1)) * 2
Step 5: Simplify the expression.
We can simplify the expression further by multiplying 2 with sin(2(2theta-1)):
- 2sin(4theta - 2)
Therefore, the derivative of g(theta) = cos^2(2theta-1) is -2sin(4theta - 2).