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Understanding the Derivative of y e^x - x^2: A Comprehensive Guide
Understanding the Derivative of y e^x - x^2: A Comprehensive Guide
Welcome to our comprehensive guide on the derivative of the function y e^x - x^2. This article aims to break down the process of finding the derivative using the principles of calculus, especially focusing on the product rule, chain rule, and basic differentiation techniques.
Introduction to Calculus and Derivatives
Calculus is a fundamental branch of mathematics, primarily concerned with the study of change. One of the key operations in calculus is the process of differentiation, which allows us to find the rate of change of a function. In this article, we will explore how to find the derivative of the function y e^x - x^2 step-by-step.
Step-by-Step Derivation of y e^x - x^2
Let's consider the function y e^x - x^2. To find its derivative, we can break this function into two parts and differentiate each part separately using the rules of differentiation.
Using the Product Rule
First, let's consider the function y e^x - x^2 by breaking it into u e^x and v 1 - x^2. We will use the product rule, which states that if we have a function in the form uv, then its derivative is given by:
d/dx (uv) u dv/dx v du/dx
Applying this rule:
d/dx (e^x (1 - x^2)) (1 - x^2) d/dx (e^x) e^x d/dx (1 - x^2)
Using the Chain Rule
Next, we need to differentiate e^x. The chain rule is used here, which states that if y e^u and u x, then the derivative is given by:
d/dx (e^x) d/dx (e^u) du/dx e^u du/dx e^x (1) e^x
Similarly, for v 1 - x^2, we use the power rule, which states that if y x^n, then the derivative is:
d/dx (x^n) n x^(n-1)
So, for 1 - x^2, we get:
d/dx (1 - x^2) 0 - 2x -2x
Combining the Results
Now, combining the results from the product rule and the chain rule, we get:
d/dx (e^x (1 - x^2)) (1 - x^2) e^x e^x (-2x) e^x (1 - x^2 - 2x)
Therefore, the derivative of y e^x - x^2 is:
y' e^x (1 - x^2 - 2x)
Conclusion
This comprehensive guide has provided you with a step-by-step understanding of how to find the derivative of the function y e^x - x^2. By using the product rule and the chain rule, we were able to break down the function into simpler parts and find its derivative accurately.