Final answer:
To find the derivatives of polar and parametric equations, we can use the chain rule. For polar equations, we find the derivative of r with respect to theta by finding dr/dt and dt/dtheta. For parametric equations, we find the derivative of y with respect to x by dividing dy/dt by dx/dt.
Step-by-step explanation:
Derivatives of Polar Equations
To find the derivative of a polar equation, we can use the chain rule. Let's say we have a polar equation given by r = f(theta), where r represents the distance from the origin to a point and theta represents the angle. To find dr/dtheta, the derivative of r with respect to theta, we can use the following formula:
Find dr/dt, the derivative of r with respect to time.
Find dt/dtheta, the derivative of t with respect to theta.
Multiply dr/dt and dt/dtheta to get the final answer.
Derivatives of Parametric Equations
To find the derivatives of parametric equations x = f(t) and y = g(t), we can use the chain rule as well. The derivatives dx/dt and dy/dt represent the rates of change of x and y with respect to t. To find dy/dx, the derivative of y with respect to x, we can use the formula:
Find dy/dt and dx/dt.
Divide dy/dt by dx/dt to get dy/dx, the final answer.