Derivatives

In Exercises 27–32, find dp/dq.

p = (sin q + cos q) / cos q

Guida verificata passo dopo passo

Step 1: Start by identifying the function p in terms of q. Here, p is given as \( p = \frac{\sin q + \cos q}{\cos q} \).

Step 2: Simplify the expression for p. Divide each term in the numerator by the denominator: \( p = \frac{\sin q}{\cos q} + \frac{\cos q}{\cos q} \). This simplifies to \( p = \tan q + 1 \).

Step 3: Differentiate p with respect to q. The derivative of \( \tan q \) with respect to q is \( \sec^2 q \), and the derivative of a constant (1) is 0.

Step 4: Combine the derivatives to find \( \frac{dp}{dq} \). Since \( p = \tan q + 1 \), \( \frac{dp}{dq} = \sec^2 q + 0 \).

Step 5: Conclude that the derivative of p with respect to q is \( \frac{dp}{dq} = \sec^2 q \).

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