📚 Derivada - Definição e Cálculo - Cálculo 1 (#16)

Introduction

This tutorial provides a clear and concise guide to understanding and calculating derivatives, a fundamental concept in calculus. We will explore the geometric interpretation of derivatives and the steps involved in calculating them, which are essential skills for success in calculus.

Step 1: Understanding the Derivative

• The derivative represents the rate of change of a function at a given point.
• Geometrically, it is interpreted as the slope of the tangent line to the curve of the function at that point.
• A positive derivative indicates that the function is increasing, while a negative derivative shows that the function is decreasing.

Key Concepts to Understand

• Function: A relation between a set of inputs and outputs.
• Tangent Line: A straight line that touches the curve at a single point without crossing it.
• Slope: The steepness of a line, calculated as the rise over run.

Step 2: Calculating the Derivative

To calculate the derivative of a function, follow these steps:

1. Identify the Function: Start with a given function ( f(x) ).

2. Use the Limit Definition: [ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} ]

   • This formula calculates the slope of the tangent line by considering the change in the function value as ( h ) approaches zero.

3. Evaluate the Limit:

   • Substitute ( f(x+h) ) and ( f(x) ) into the limit formula.
   • Simplify the expression and compute the limit as ( h ) approaches zero.

Example Calculation

For the function ( f(x) = x^2 ):

1. Substitute into the limit: [ f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} ]
2. Expand and simplify: [ = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h} ] [ = \lim_{h \to 0} (2x + h) = 2x ]

Step 3: Applying Rules of Differentiation

Familiarize yourself with basic differentiation rules to simplify calculations:

• Power Rule: If ( f(x) = x^n ), then ( f'(x) = nx^{n-1} ).
• Sum Rule: If ( f(x) = g(x) + h(x) ), then ( f'(x) = g'(x) + h'(x) ).
• Product Rule: If ( f(x) = g(x) \cdot h(x) ), then ( f'(x) = g'(x)h(x) + g(x)h'(x) ).
• Quotient Rule: If ( f(x) = \frac{g(x)}{h(x)} ), then ( f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} ).

Conclusion

Understanding and calculating derivatives is crucial in calculus, as it provides insights into the behavior of functions. By mastering the limit definition and differentiation rules, you can analyze and interpret functions more effectively. As a next step, practice calculating derivatives using various functions and apply these concepts to real-world problems, such as physics and engineering scenarios where rates of change are essential.