📚 Topic Summary
The quotient rule in calculus is essential for finding the derivative of a function that is expressed as a ratio of two other functions. For trigonometric functions like secant, cosecant, tangent, and cotangent, we can use the quotient rule along with known derivatives of sine and cosine to derive their respective derivative formulas. This activity focuses on applying the quotient rule to understand how these trigonometric derivatives are obtained.
🧮 Part A: Vocabulary
Match the term with its correct definition:
| Term |
Definition |
| 1. Quotient Rule |
A. $\frac{-d}{dx} \csc(x) = -\csc(x)\cot(x)$ |
| 2. $\frac{d}{dx} \sec(x)$ |
B. A rule to find the derivative of a function which is the ratio of two functions. |
| 3. $\frac{d}{dx} \tan(x)$ |
C. $\sec^2(x)$ |
| 4. $\frac{d}{dx} \cot(x)$ |
D. $\frac{d}{dx} \sec(x) = \sec(x)\tan(x)$ |
| 5. $\frac{d}{dx} \csc(x)$ |
E. $-\csc^2(x)$ |
✍️ Part B: Fill in the Blanks
To derive the derivative of $\tan(x)$ using the quotient rule, we can express $\tan(x)$ as $\frac{\sin(x)}{\cos(x)}$. Applying the quotient rule, we get $\frac{d}{dx} \tan(x) = \frac{\cos(x) \cdot \frac{d}{dx} \sin(x) - \sin(x) \cdot \frac{d}{dx} \cos(x)}{(\cos(x))^2}$. This simplifies to $\frac{\cos(x) \cdot \cos(x) - \sin(x) \cdot (-\sin(x))}{\cos^2(x)} = \frac{\cos^2(x) + \sin^2(x)}{\cos^2(x)}$. Since $\cos^2(x) + \sin^2(x) = 1$, we have $\frac{1}{\cos^2(x)}$, which is equal to ____, or $\sec^2(x)$. Similarly, we can derive the other trigonometric derivatives using the quotient rule.
🤔 Part C: Critical Thinking
Explain why it is useful to know the quotient rule when finding derivatives of trigonometric functions even though other methods exist.