Introduction
Understanding the basics of derivatives is essential for anyone starting their journey in calculus. In this blog, we’ll explore the derivatives of some fundamental functions: x, 1/x, and x raised to a power (x^a). These functions form the foundation of many more complex calculus problems, and mastering their derivatives will help you tackle more advanced equations with confidence.
Derivatives of x
Let’s start with the simplest function: x. The derivative of x with respect to x is simply 1. This is because when you change x by a small amount, the function changes at a constant rate of 1.
Example:
If f(x) = x, then f’(x) = 1.
Derivatives of 1/x
The next function we’ll differentiate is 1/x. We can rewrite 1/x as x^(-1). Now, applying the power rule, we get:
• The derivative of x^(-1) is -1 * x^(-2), or simply -1/x².
Example:
If f(x) = 1/x, then f’(x) = -1/x².
Derivatives of x^a (Power Rule)
The power rule is one of the most useful rules in calculus. It tells us that the derivative of x raised to any power (x^a) is a * x^(a - 1).
Example:
If f(x) = x³, then f’(x) = 3x².
Similarly, if f(x) = x^(1/2), then f’(x) = (1/2) * x^(-1/2).
Conclusion
By mastering these basic derivatives, you’ll have a solid foundation to tackle more advanced calculus problems. Whether you’re solving real-world optimization problems or diving deeper into math, understanding the derivatives of x, 1/x, and x^a is key to your success.
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