What Is The Derivative Of 3x

What is the Derivative of 3x? A Simple Explanation

This article will clearly and concisely explain how to find the derivative of 3x, a fundamental concept in calculus. Understanding this will lay a solid foundation for tackling more complex derivatives. We'll cover the power rule, a crucial tool for differentiation, and provide examples to solidify your understanding.

The derivative of a function represents its instantaneous rate of change. In simpler terms, it tells us how much the function's output changes for a tiny change in its input. This is extremely useful in various fields, from physics (calculating velocity and acceleration) to economics (analyzing marginal costs).

Understanding the Power Rule

The power rule is a shortcut for finding the derivative of functions in the form xⁿ, where 'n' is a constant. The rule states:

d/dx (xⁿ) = nx^n-1

This means you multiply the function by the exponent and then decrease the exponent by 1.

Applying the Power Rule to 3x

The function 3x can be rewritten as 3x¹. Now, let's apply the power rule:

1. Identify n: In our case, n = 1.
2. Apply the formula: d/dx (3x¹) = 3 * 1 * x^1-1
3. Simplify: This simplifies to 3 * x⁰. Remember that any number raised to the power of 0 is 1 (except 0⁰ which is undefined).
4. Final Result: Therefore, the derivative of 3x is 3.

Visualizing the Derivative

Imagine the graph of y = 3x. It's a straight line with a slope of 3. The derivative, being the instantaneous rate of change, represents the slope of this line at any point. Since the slope is constant, the derivative is a constant value: 3.

Examples with Similar Functions

Let's extend our understanding with some similar examples:

• Derivative of 5x: Using the power rule, d/dx (5x) = 5.
• Derivative of -2x: d/dx (-2x) = -2.
• Derivative of 10x²: Applying the power rule, d/dx (10x²) = 20x.

This illustrates how the constant multiplier in front of the 'x' term simply carries through the differentiation process.

Conclusion

The derivative of 3x is simply 3. Understanding the power rule is key to mastering differentiation and opens the door to solving more complex calculus problems. Remember, practice makes perfect! Work through several examples to solidify your grasp of this fundamental concept. By understanding this basic principle, you’ll be well-equipped to explore the fascinating world of calculus and its applications.