Finding the Derivative of x²-2 at x=10

At its core, a derivative represents the instantaneous rate of change of a function. For our subject, x²-2, the derivative calculates the slope of the tangent line to its graph at a given point. This concept forms the cornerstone of calculus, empowering us to study the intricate behaviors of functions.

To ascertain the numerical value of the derivative, we employ the power rule, a mathematical formula specifically tailored for differentiating polynomial functions. In our case, the power rule dictates that the derivative of x² is 2x, and since -2 is a constant, its derivative is 0. Therefore, the derivative of x²-2 is simply 2x.

Evaluating at x=10: Precision in Calculation

Having established the general form of the derivative, we now seek to determine its value at the specific point x=10. Plugging this value into our derived expression, we obtain 2(10) = 20. This result encapsulates the notion that the slope of the tangent line to the graph of x²-2 at x=10 is 20.

Applications of the Derivative: A Spectrum of Utility

The derivative of x²-2, and derivatives in general, find far-reaching applications in diverse scientific domains. Below, we explore two notable examples:

• Optimization: In fields like engineering and economics, optimization models hinge upon the concept of derivatives to determine optimal solutions. The derivative provides invaluable insights into the maxima and minima of functions, guiding decision-making towards optimal outcomes.
• Physics: Derivatives play a pivotal role in physics, enabling scientists to analyze the motion of objects. From characterizing velocity and acceleration to determining the trajectories of projectiles, derivatives offer a powerful toolset for deciphering the dynamics of our physical world.

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Find The Derivative Of X 2 2 At X 10

Conclusion

The derivative of x²-2 at x=10 stands as a crucial concept in mathematics, encompassing both theoretical significance and extensive practical applications. Understanding this derivative empowers us to delve into complex mathematical problems, unravel the underlying mechanisms of real-world phenomena, and leverage the transformative power of mathematics to expand our knowledge and shape our world. As you continue your mathematical journey, may this newfound knowledge serve as a beacon, guiding you towards new discoveries and a deeper understanding of the mathematical universe.