F5 Add Math | Basic Differentiation | Asas Pembezaan

Introduction

This tutorial will guide you through the basics of differentiation, a fundamental concept in calculus that allows you to find the rate at which a function is changing. Understanding differentiation is essential for various applications in mathematics, physics, and engineering. By the end of this guide, you'll be equipped with the essential rules of differentiation and practical tips for solving related problems.

Step 1: Understand the Power Rule

The power rule is a key principle in differentiation. It states that if you have a function in the form of ( f(x) = x^n ), the derivative ( f'(x) ) is calculated as follows:

• Multiply the exponent ( n ) by the coefficient (which is 1 if not specified).
• Decrease the exponent by 1.

Example

For the function ( f(x) = x^3 ):

• The derivative is calculated as:
  • ( f'(x) = 3x^{3-1} = 3x^2 )

Step 2: Differentiate Constant Functions

When differentiating constant functions, remember that the derivative of any constant is zero.

Example

For the function ( f(x) = 5 ):

• The derivative is:
  • ( f'(x) = 0 )

Step 3: Apply the Sum Rule

When you have a function that is the sum of multiple terms, differentiate each term separately.

Example

For the function ( f(x) = x^2 + 3x + 5 ):

• Differentiate each term:
  • ( f'(x) = 2x + 3 + 0 = 2x + 3 )

Step 4: Work with Negative Exponents

If a function involves negative exponents, apply the power rule as usual.

Example

For the function ( f(x) = x^{-2} ):

• The derivative is:
  • ( f'(x) = -2x^{-3} )

Step 5: Differentiate with Product Rule

When differentiating functions that are products of two functions, use the product rule, which states that if ( f(x) = g(x) \cdot h(x) ), then:

• ( f'(x) = g'(x)h(x) + g(x)h'(x) )

Example

For the function ( f(x) = x^2 \cdot \sin(x) ):

• Differentiate using the product rule:
  • ( f'(x) = 2x \cdot \sin(x) + x^2 \cdot \cos(x) )

Step 6: Differentiate with Quotient Rule

For functions that are quotients, use the quotient rule which states that if ( f(x) = \frac{g(x)}{h(x)} ):

• ( f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} )

Example

For the function ( f(x) = \frac{x^2}{x+1} ):

• Differentiate using the quotient rule:
  • ( f'(x) = \frac{(2x)(x+1) - (x^2)(1)}{(x+1)^2} )

Conclusion

Differentiation is a powerful tool in mathematics that helps you understand how functions behave. Remember the key rules: the power rule, sum rule, product rule, and quotient rule. Practice applying these rules to different functions to build your skills. As you advance, explore more complex topics in calculus, such as higher-order derivatives and applications of differentiation in real-world scenarios. Continue practicing, and you'll become proficient in differentiation!