To find the first and second derivatives of the function y = 5 sin(t) + 6 cos(t) with respect to t, we can follow these steps:
Step 1: Finding the First Derivative (dydx)
The first derivative of y with respect to t, denoted as dydx, can be calculated using the rules of differentiation.
1. Differentiate 5 sin(t): The derivative of sin(t) is cos(t), so:
- rac{d}{dt}[5 sin(t)] = 5 cos(t)
2. Differentiate 6 cos(t): The derivative of cos(t) is -sin(t), so:
- rac{d}{dt}[6 cos(t)] = -6 sin(t)
Putting it all together, we have:
dydx = 5 cos(t) – 6 sin(t)
Step 2: Finding the Second Derivative (d2ydx2)
The second derivative, denoted as d2ydx2, is found by differentiating the first derivative.
1. Differentiate 5 cos(t): The derivative of cos(t) is -sin(t), so:
- rac{d}{dt}[5 cos(t)] = -5 sin(t)
2. Differentiate -6 sin(t): The derivative of sin(t) is cos(t), so:
- rac{d}{dt}[-6 sin(t)] = -6 cos(t)
Combining these results gives us:
d2ydx2 = -5 sin(t) – 6 cos(t)
Summary
In summary, the first and second derivatives of the function are:
- dydx = 5 cos(t) – 6 sin(t)
- d2ydx2 = -5 sin(t) – 6 cos(t)
This process demonstrates the straightforward application of differentiation rules to find the derivatives of trigonometric functions.