To find the first and second derivatives of the function y = cos²(t), we will use the chain rule for differentiation.
Step 1: Finding dy/dx
First, we need to differentiate y with respect to t:
Using the chain rule:
dy/dt = d(cos²(t))/dt
We apply the derivative of cos²(t):
dy/dt = 2cos(t) * (-sin(t)) = -2cos(t)sin(t)
Now, we can express dy/dx using the chain rule along with dx/dt. Since dx/dt = 1, we have:
dy/dx = dy/dt = -2cos(t)sin(t)
Step 2: Finding d²y/dx²
Next, we need to find the second derivative, d²y/dx²:
To do that, we differentiate dy/dx with respect to t:
d²y/dt² = d(-2cos(t)sin(t))/dt
Using the product rule:
d²y/dt² = -2[cos(t) * d(sin(t))/dt + sin(t) * d(cos(t))/dt]
This expands to:
d²y/dt² = -2[cos(t)(cos(t)) + sin(t)(-sin(t))] = -2[cos²(t) – sin²(t)]
Now, using the chain rule to express d²y/dx², we have:
d²y/dx² = d²y/dt² / (dx/dt)² = -2[cos²(t) – sin²(t)]
Final Results
Thus, we have:
- dy/dx = -2cos(t)sin(t)
- d²y/dx² = -2[cos²(t) – sin²(t)]
These results tell us how y changes with t and give insights into the curvature of the function within the specified interval [0, π].