The differential coefficient is essentially another term for the derivative of a function. It indicates the rate at which the function value, usually denoted \(y\), changes with respect to a change in the variable \(x\). In the example \(y = 4x^2 + 5x - 3\), the differential coefficient is found to be \(\frac{dy}{dx} = 8x + 5\).
This means that for any small change in \(x\), the change in \(y\) can be predicted by multiplying this change by the value of the differential coefficient:
- If \(\Delta x\) is a small change, then the approximate change in \(y\) is \(\Delta y = \frac{dy}{dx} \cdot \Delta x\).
Understanding differential coefficients helps in analyzing and evaluating how functions behave under different conditions and is integral to calculus and various fields requiring modeling and prediction.