Final answer:
The derivative of the function r = (secθ + tanθ) - 4 is tanθ - sec^2θ + 1.
Step-by-step explanation:
To find the derivative of the function r = (secθ + tanθ) - 4, we can use the chain rule. Let's start by rewriting the function in terms of sine and cosine:
r = (1/cosθ + sinθ/cosθ) - 4
r = (1 + sinθ) / cosθ - 4
Now, we can find the derivative:
r' = [(cosθ)(0) - (1 + sinθ)(-sinθ)] / (cosθ)^2
r' = sinθ / cosθ - sin^2θ / cos^2θ
Using the Pythagorean identity sin^2θ = 1 - cos^2θ, we can simplify:
r' = sinθ / cosθ - (1 - cos^2θ) / cos^2θ
r' = sinθ / cosθ - 1 / cos^2θ + cos^2θ / cos^2θ
r' = tanθ - sec^2θ + 1