Questions: Calculate the derivative of the following function. y=cos(20t+1) dy/dt=□

Solution Steps

To find the derivative of the function \( y = \cos(20t + 1) \), we will use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Here, the outer function is \(\cos(u)\) and the inner function is \(u = 20t + 1\).

We start with the function given in the problem: \[ y = \cos(20t + 1) \]

To find the derivative \(\frac{dy}{dt}\), we apply the chain rule. The derivative of \(\cos(u)\) is \(-\sin(u)\), where \(u = 20t + 1\). Thus, we have: \[ \frac{dy}{dt} = -\sin(20t + 1) \cdot \frac{du}{dt} \] where \(\frac{du}{dt} = 20\).

Substituting \(\frac{du}{dt}\) into the derivative expression, we get: \[ \frac{dy}{dt} = -\sin(20t + 1) \cdot 20 = -20\sin(20t + 1) \]

Final Answer