Solved examples: Applying the Chain Rule to trigonometric functions

📚 Quick Study Guide

• 📐 The Chain Rule: If $y = f(g(x))$, then $\frac{dy}{dx} = \frac{dy}{dg} \cdot \frac{dg}{dx}$.
• 🍎 Trigonometric Derivatives: Remember these key derivatives:
  • $\frac{d}{dx}(\sin x) = \cos x$
  • $\frac{d}{dx}(\cos x) = -\sin x$
  • $\frac{d}{dx}(\tan x) = \sec^2 x$
  • $\frac{d}{dx}(\csc x) = -\csc x \cot x$
  • $\frac{d}{dx}(\sec x) = \sec x \tan x$
  • $\frac{d}{dx}(\cot x) = -\csc^2 x$
• 💡 Applying the Chain Rule to Trigonometric Functions: Combine the chain rule with trigonometric derivatives. For example, if $y = \sin(x^2)$, then $\frac{dy}{dx} = \cos(x^2) \cdot 2x$.
• 📝 General Steps:
  1. Identify the 'outer' and 'inner' functions.
  2. Differentiate the outer function, keeping the inner function as is.
  3. Multiply by the derivative of the inner function.

🧪 Practice Quiz

1. What is the derivative of $y = \sin(3x)$?

   1. $3\cos(3x)$
   2. $\cos(3x)$
   3. $-3\cos(3x)$
   4. $-\cos(3x)$

2. Find $\frac{dy}{dx}$ if $y = \cos(x^2 + 1)$.

   1. $-2x\sin(x^2 + 1)$
   2. $2x\sin(x^2 + 1)$
   3. $-\sin(x^2 + 1)$
   4. $\sin(x^2 + 1)$

3. Differentiate $y = \tan(2x)$.

   1. $2\sec^2(2x)$
   2. $\sec^2(2x)$
   3. $2\cot(2x)$
   4. $\cot(2x)$

4. What is the derivative of $y = \csc(5x)$?

   1. $-5\csc(5x)\cot(5x)$
   2. $5\csc(5x)\cot(5x)$
   3. $-\csc(5x)\cot(5x)$
   4. $\csc(5x)\cot(5x)$

5. Find $\frac{dy}{dx}$ if $y = \sec(x^3)$.

   1. $3x^2\sec(x^3)\tan(x^3)$
   2. $\sec(x^3)\tan(x^3)$
   3. $3x^2\tan(x^3)$
   4. $\sec(3x^2)\tan(3x^2)$

6. Differentiate $y = \cot(4x)$.

   1. $-4\csc^2(4x)$
   2. $4\csc^2(4x)$
   3. $-\csc^2(4x)$
   4. $\csc^2(4x)$

7. What is the derivative of $y = \sin^2(x)$?

   1. $2\sin(x)\cos(x)$
   2. $\sin(x)\cos(x)$
   3. $2\cos(x)$
   4. $\cos^2(x)$