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The concept of percentage error is a measure of how much an estimated value differs from the true value, expressed as a percentage. It helps understand the accuracy of a measurement or calculation. For a variable such as \( x \), if you know the measurement error, you can find the percentage error through the formula:

• Percentage Error = \( \left( \frac{\text{error}}{\text{true value}} \right) \times 100\% \)

In practice, this means if you know there's a certain percentage error in \( x \), like \( \pm 15\% \) as in our exercise, you can infer what effect that error has on \( y \), which is affected by \( x \). The steps shown in the solution help translate the known error in \( x \) into an estimated error in \( y \). This is crucial in fields like engineering and physical sciences, where precision is key.

Differentiation is a fundamental concept in calculus that deals with understanding rates of change. It is the process of finding the derivative of a function, which gives you an expression that describes how the function changes at any point.

• The notation for differentiation is \( \frac{dy}{dx} \), which reads "the derivative of \( y \) with respect to \( x \)".

In our exercise, by differentiating the function \( y = 4x^3 \), we arrive at the derivative \( \frac{dy}{dx} = 12x^2 \). This derivative tells us how the value of \( y \) changes as \( x \) changes, which is essential in error analysis to see how deviations in \( x \) directly influence \( y \). It essentially lays the groundwork for understanding how sensitive a function is to changes in its input.

Differentials provide a way to approximate how changes in one variable can affect another. Once we have the derivative, differentials help to calculate the small change in the dependent variable given a change in the independent variable.

• For our function, if \( y = 4x^3 \) and \( dy = 12x^2 dx \), \( dy \) represents the change in \( y \) for a small change \( dx \) in \( x \).

This connection means that if \( x \) changes by a small amount, \( dx \), then \( y \) will change by \( dy \). The equation \( dy = 12x^2 dx \) specifically helps us quantify how much this change in \( y \) will be. It essentially captures the ripple effect from altering \( x \) over to \( y \). This understanding is crucial when analyzing errors in measurements or predictions.

Mathematical derivation involves using known mathematical principles to arrive at a new result or formula. It's about step-by-step reasoning from what we know to what we need to find out.

• In the problem at hand, we use derivation to link the error in \( x \) to the error in \( y \).
• This involves using the derivative \( \frac{dy}{dx} \) to express \( dy \) in terms of \( dx \) and then relating \( dy \) to \( y \) and \( dx \) to \( x \).

Through simple algebraic manipulation and substitution, such as \( \frac{dy}{y} = \frac{3dx}{x} = 3 \times 0.15 = \pm 0.45 \), we derive the percentage error in \( y \) from the percentage error in \( x \). This process highlights the importance of a structured approach in mathematics to solve complex problems by breaking them down into simpler, manageable parts.