Differential Calculus: Finding dy/dx Given x and dy/dt

In the study of differential calculus, understanding the relationship between different rates of change, such as dy/dx and dy/dt, is a foundational concept. This article provides a step-by-step guide on how to find dy/dx given the expressions x 2t^3 and dy/dt 14t^3.

Introduction to the Problem

Given the equations:

x 2t^3 (Equation 1) dy/dt 14t^3 (Equation 2)

The task is to find the relationship between dy/dx in terms of x.

Step-by-Step Solution

1. Differentiating x with respect to t

To find dx/dt, we differentiate x 2t^3 with respect to t using the power rule:

dx/dt  d/dt(2t^3)  2*3*t^2  6t^2

Therefore:

dx/dt 6t^2

2. Finding dy/dx using the Chain Rule

The chain rule states that:

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

Substituting the given values:

dy/dx (14t^3) / (6t^2) (14/6) * t (7/3) * t

However, we can further simplify this expression to:

dy/dx (7/6) * t

3. Expressing the Final Answer in Terms of x

To express dy/dx in terms of x, we need to solve for t in terms of x. From Equation 1:

x 2t^3 implies t^3 x/2

Taking the cube root of both sides:

t (x/2)^(1/3) (x/2)^(1/3) x^(1/3) / 2^(1/3)

Substituting this back into our expression for dy/dx:

dy/dx (7/6) * (x^(1/3) / 2^(1/3))

4. Simplifying the Expression

We can further simplify this expression:

dy/dx (7/6) * (x/2)^(1/3) (7/6) * (x/2)^(1/3) (7/6) * x^(1/3) * 2^(-1/3)

Which simplifies to:

dy/dx (7/6) * (1/2)^(1/3) * x^(1/3) (7/6) * (1/2)^(1/3) * x^(1/3) (7/12) * x^(1/3)

Therefore, the final expression for dy/dx in terms of x is:

dy/dx (7/12) * x^(1/3)

Another Method

An alternative method involves directly substituting the given values and simplifying:

dy/dx (dy/dt) / (dx/dt) (14t^3) / (6t^2) (14/6) * t (7/3) * t

Substituting t (x/2)^(1/3):

dy/dx (7/3) * (x/2)^(1/3)

Conclusion

In conclusion, given the equations x 2t^3 and dy/dt 14t^3, the relationship between dy/dx in terms of x is:

dy/dx (7/12) * x^(1/3)

This method showcases the application of basic calculus principles in solving real-world problems, emphasizing the importance of understanding the chain rule and substitution techniques.