Final Answer:
The partial derivative of x with respect to x for the equation x²+4y²+3y²=2 is 2x.
Step-by-step explanation:
To find the partial derivative of x with respect to x, we hold y as a constant and differentiate the equation with respect to x. Using the chain rule, we can write:
∂(x²+4y²+3y²)/∂x = 2x + 0 (since y is held constant)
Simplifying, we get:
∂x/∂x = 2x
This derivative represents how the value of x changes with respect to x, while keeping y constant. It is useful in finding local linear approximations of functions and in optimization problems where we want to find the maximum or minimum value of a function by finding its stationary points. In this case, the partial derivative tells us that the rate of change of x with respect to x is directly proportional to the value of x itself. This means that if we increase the value of x by a small amount, then the value of x² will also increase by a factor of 2x. This is because the derivative measures the slope of the curve at a given point, and in this case, it is a straight line passing through that point with a slope of 2x.
Therefore, if we move along this line in the direction of increasing x, then our position on the curve will also increase at a rate proportional to our current position on the curve. This concept is important in many fields, such as physics, engineering, and economics, where it helps us understand how variables are related and how they change over time. By calculating partial derivatives, we can gain insights into complex systems and make more informed decisions based on our analysis.