Matematika kelas XII - Turunan Fungsi ln, Fungsi e, Fungsi Eksponen dan Fungsi Logaritma - 2022

Introduction

In this tutorial, we will explore the concepts of derivatives related to logarithmic functions, exponential functions, and the natural logarithm. This guide is based on a comprehensive video lesson that covers essential formulas and problem-solving techniques in calculus, specifically for students in grade XII. Understanding these derivatives is crucial for mastering higher-level mathematics and applications in various fields.

Step 1: Understanding Natural Logarithm Derivatives

The derivative of the natural logarithm function is fundamental in calculus. Here’s how to differentiate it:

• If you have a function ( f(x) = \ln(x) ), the derivative is: [ f'(x) = \frac{1}{x} ]
• Practical Tip: Ensure ( x > 0 ) since the natural logarithm is undefined for zero and negative numbers.

Step 2: Derivatives of Exponential Functions

Exponential functions are another key area in calculus. The derivative of the exponential function is straightforward:

• If ( f(x) = e^x ), then the derivative is: [ f'(x) = e^x ]
• If ( f(x) = a^x ) (where ( a ) is a constant), the derivative is: [ f'(x) = a^x \ln(a) ]
• Common Pitfall: Remember that the base ( e ) (approximately 2.718) has a unique property where its derivative remains the same.

Step 3: Derivatives of Logarithmic Functions with Different Bases

For logarithmic functions with bases other than ( e ):

• If ( f(x) = \log_a(x) ), the derivative is: [ f'(x) = \frac{1}{x \ln(a)} ]
• Practical Advice: Always check the base of the logarithm when calculating the derivative.

Step 4: Applying the Chain Rule

When dealing with composite functions, the chain rule is essential:

• If ( f(x) = \ln(g(x)) ), then the derivative using the chain rule is: [ f'(x) = \frac{g'(x)}{g(x)} ]
• Example: If ( g(x) = x^2 + 1 ), then: [ f'(x) = \frac{2x}{x^2 + 1} ]

Step 5: Practice Problems

To solidify your understanding, solve these practice problems:

1. Find the derivative of ( f(x) = \ln(3x + 2) ).
2. Calculate the derivative of ( f(x) = 5^x ).
3. Differentiate ( f(x) = \log_2(x^3 + 1) ).

Conclusion

In this tutorial, we covered the derivatives of natural logarithm, exponential functions, and logarithmic functions with different bases. We also discussed the chain rule and provided practice problems to enhance your understanding. Mastering these concepts will not only help you in your current mathematics course but also lay a strong foundation for future studies in calculus and beyond. For further practice, revisit the provided video and explore additional examples.