Solving for dy/dx When x1 in the Equation y x^2 1^2 - 3x

In this article, we explore the problem of finding the derivative of the equation y x^2 1^2 - 3x at the specific point where x 1. This involves a step-by-step application of calculus, particularly differential calculus. We will derive the expression for dy/dx and evaluate it at the given value of x.

Introduction to Differential Calculus

Differential calculus is a fundamental branch of mathematics that focuses on the study of derivative and tangent lines. A derivative of a function provides the rate at which the function is changing at any given point, making it an essential tool in various fields, including physics, engineering, and economics. In this context, we aim to find the derivative of the function y x^2 1^2 - 3x with respect to x and then evaluate it at x 1.

Step-by-Step Derivation

Starting Expression: The given equation is y x^2 1^2 - 3x. Substitute and Simplify: When x 1, substitute the value into the equation to find y.

y (1)^2 (1)^2 - 3(1) 1 1 - 3 -1.

Logarithmic Transformation: To simplify finding the derivative, we take the natural logarithm of both sides of the original equation.

y x^2 1^2 - 3x

ln(y) ln(x^2 1) - 3ln(x)

Apply Implicit Differentiation: Differentiate both sides with respect to x.

y' / y d/dx[ln(x^2 1)] - 3d/dx[ln(x)]

y' / y 2x / (x^2 1) - 3 / x

Substitute y and Simplify: Substitute y x^2 1 - 3x for the y' expression.

y' (x^2 1 - 3x) * [2x / (x^2 1) - 3 / x]

Evaluating at x 1

Now we evaluate the expression for the derivative at x 1.

Substitute x 1:

dy/dx (1^2 1 - 3) * [2(1) / (1^2 1) - 3 / 1]

dy/dx (1 1 - 3) * [2 / 2 - 3]

dy/dx (-1) * [1 - 3] -1 * (-2) 2.

Additional Techniques and Simplifications

Logarithmic Simplification: An alternative method is to simplify the expression using logarithmic properties.

dy/dx (1^2 1 - 3) * [3ln2 - 2 / 2] -0.5 * [3ln2 - 1] -0.5 * (1.5ln2 - 1) -0.75ln2 0.5.

Alternatively, we can express it as:

dy/dx 0.5 * (ln8 - 1)

where, ln8 3ln2.

Conclusion

In conclusion, the derivative of the equation y x^2 1^2 - 3x evaluated at x 1 is 2. This result provides insight into the rate of change of the function at the specified point, demonstrating the power and utility of differential calculus.