To find the gradient of a curve at a specific point, we need to use the fundamental concept of differentiation. The gradient of a curve at any point is determined by its first derivative, \(\frac{dy}{dx}\). This derivative provides the instantaneous rate of change of \(y\) with respect to \(x\), which is equivalent to the slope of the tangent line to the curve at that particular point.

Gradient Definition

The gradient of a curve visually represents its steepness or inclination at any given location along its path. For a mathematical function expressed as \(y = f(x)\), the general formula for its gradient is obtained by calculating its derivative with respect to \(x\), commonly denoted as \(\frac{dy}{dx}\).

Curve Equation Analysis

The provided equation for the curve is:

\(y = 3x^2 - 7x + 2\)

To derive the general expression for the gradient, we must differentiate this equation term by term with respect to \(x\). The key rules of differentiation applicable here are:

• The power rule for differentiation states that for a term in the form \(ax^n\), its derivative is \(anx^{n-1}\).
• The derivative of a constant term (a number without any \(x\)) is always zero.

Differentiation Process

Let's apply the differentiation rules to each term of the curve equation:

1. Differentiate \(3x^2\): Applying the power rule, \(\frac{d}{dx}(3x^2) = 3 \times 2x^{2-1} = 6x\).
2. Differentiate \(-7x\): Since \(x\) can be thought of as \(x^1\), applying the power rule gives \(\frac{d}{dx}(-7x) = -7 \times 1x^{1-1} = -7x^0\). As \(x^0 = 1\), this simplifies to \(-7\).
3. Differentiate \(2\): As 2 is a constant, its derivative is \(\frac{d}{dx}(2) = 0\).

Combining the derivatives of these individual terms, the overall derivative of the curve, \(\frac{dy}{dx}\), which gives the general gradient function, is:

\(\frac{dy}{dx} = 6x - 7\)

Gradient Evaluation at the Specific Point

The final step is to find the exact gradient at the given specific point \((1, -2)\). To do this, we substitute the \(x\)-coordinate of this point (which is \(x = 1\)) into our derived general gradient function \(\frac{dy}{dx} = 6x - 7\).

Substitute \(x = 1\) into the derivative:

\(\text{Gradient} = 6(1) - 7\)

Perform the multiplication and subtraction:

\(\text{Gradient} = 6 - 7\)

\(\text{Gradient} = -1\)

Thus, the gradient of the curve \(y = 3x^2 - 7x + 2\) at the specified point \((1, -2)\) is \(-1\).