Turunan Aturan Rantai Trigonometri - Rumus Cepat

Introduction

This tutorial focuses on the chain rule for derivatives in trigonometry, providing a quick and efficient way to understand and apply this important concept. The chain rule is a fundamental technique in calculus that allows you to differentiate composite functions, which is especially useful when dealing with trigonometric functions.

Step 1: Understanding the Chain Rule Concept

• The chain rule states that if you have a composite function, say ( h(x) = f(g(x)) ), the derivative ( h'(x) ) can be expressed as:

  [ h'(x) = f'(g(x)) \cdot g'(x) ]

• Here’s what each part represents:

  • ( f ) is the outer function.
  • ( g ) is the inner function.
  • ( f' ) is the derivative of the outer function evaluated at the inner function ( g(x) ).
  • ( g' ) is the derivative of the inner function.

Practical Tip

• Always identify the inner and outer functions first when applying the chain rule.

Step 2: Applying the Chain Rule to Trigonometric Functions

1. Identify the outer and inner functions in your trigonometric expression.

   • Example: For ( h(x) = \sin(x^2) ):
     • Outer function: ( f(u) = \sin(u) )
     • Inner function: ( g(x) = x^2 )

2. Differentiate both functions:

   • Derivative of the outer function: ( f'(u) = \cos(u) )
   • Derivative of the inner function: ( g'(x) = 2x )

3. Apply the chain rule:

   • Substitute ( g(x) ) back into the derivative of the outer function:

   [ h'(x) = \cos(g(x)) \cdot g'(x) = \cos(x^2) \cdot (2x) ]

4. Final result:

   [ h'(x) = 2x \cos(x^2) ]

Common Pitfall

• Forgetting to differentiate both the inner and outer functions can lead to incorrect results.

Step 3: Examples of Chain Rule with Different Trigonometric Functions

• Example 1: Differentiate ( h(x) = \tan(3x + 1) )

  • Outer function: ( f(u) = \tan(u) )
  • Inner function: ( g(x) = 3x + 1 )
  • Derivative:

  [ h'(x) = \sec^2(3x + 1) \cdot 3 = 3 \sec^2(3x + 1) ]

• Example 2: Differentiate ( h(x) = \cos(x^3) )

  • Outer function: ( f(u) = \cos(u) )
  • Inner function: ( g(x) = x^3 )
  • Derivative:

  [ h'(x) = -\sin(x^3) \cdot 3x^2 = -3x^2 \sin(x^3) ]

Conclusion

The chain rule is essential for differentiating composite trigonometric functions. By identifying the outer and inner functions and applying the rule correctly, you can easily find derivatives for a wide range of expressions. As you practice more examples, the process will become more intuitive. For further study, consider exploring higher-order derivatives and their applications in real-world problems.