Introduction

In this article, we will be discussing how to calculate dy/dt for the functions y x^2 - 3x 1 and x t^2 - 2. This involves the application of the chain rule, which is an essential tool in differential calculus.

Understanding the Functions

First, let's define the functions given:

y x^2 - 3x 1 - A quadratic function in terms of x.

x t^2 - 2 - A function of time t.

The Chain Rule

The chain rule is used to differentiate composite functions. It can be stated as:

dy/dt (dy/dx) * (dx/dt)

This rule is particularly useful when we need to differentiate a function with respect to a variable that is not the same as the variable in the function itself.

Solving the Problem

To find dy/dt, we first calculate dy/dx and dx/dt, and then use the chain rule to find dy/dt.

Step 1: Calculate dy/dx

We start by differentiating y x^2 - 3x 1 with respect to x:

dy/dx 2x - 3

Step 2: Calculate dx/dt

We then differentiate x t^2 - 2 with respect to t:

dx/dt 2t

Step 3: Apply the Chain Rule

Using the chain rule, we combine the results of the previous steps:

dy/dt (dy/dx) * (dx/dt) (2x - 3) * 2t

Substituting x t^2 - 2 into the expression for dy/dt:

dy/dt (2(t^2 - 2) - 3) * 2t

Step 4: Simplify the Expression

Simplifying the expression for dy/dt will give us the final result:

dy/dt (2t^2 - 4 - 3) * 2t (2t^2 - 7) * 2t 4t^3 - 14t

Hence, the final expression for dy/dt is:

dy/dt 4t^3 - 14t

Conclusion

This process of using the chain rule allows us to find the derivative of a function with respect to a different variable, which is a common operation in calculus and differential equations. By following these steps, we have successfully calculated the desired derivative.

Key Takeaways:

The chain rule is a powerful tool in calculus. Understanding the relationship between variables through differentiation is essential. Substitution and algebraic simplification are crucial in solving such problems.

For more information on related topics in calculus and differential equations, feel free to explore further resources or seek guidance from a mathematics tutor or relevant educational platform.