📚 Differentiating Quadratics | 二次函数的求导
Quadratic functions are among the simplest non-linear functions, and learning to differentiate them is a key skill in A-Level Mathematics. Understanding how to find the derivative of an expression like ax² + bx + c opens the door to analysing gradients, tangents, stationary points and optimisation problems.
二次函数是最简单的非线性函数之一，掌握它们的求导方法是 A-Level 数学中的一项关键技能。学会求形如 ax² + bx + c 的表达式的导数，能够帮助你分析梯度、切线、驻点以及优化问题。
1. Quadratic Functions and Their Graphs | 二次函数及其图像
re constants and a ≠ 0. Its graph is a parabola, which opens upwards if a > 0 and downwards if a < 0.
二次函数的一般形式为 f(x) = ax² + bx + c，其中 a、b 和 c 为常数且 a ≠ 0。其图像是一条抛物线，当 a > 0 时开口向上，当 a < 0 时开口向下。
The vertex of the parabola represents the minimum or maximum point of the function, and differentiation helps us locate this vertex precisely.
抛物线的顶点代表函数的最小值或最大值点，而微分可以帮助我们准确地找到该顶点。
2. Understanding the Gradient of a Curve | 理解曲线的梯度
For a straight line, the gradient is constant. For a quadratic curve, the gradient changes at every point. Differentiation allows us to find the gradient function, which gives the gradient at any point on the curve.
对于直线，梯度是恒定的。对于二次曲线，梯度在每一点都在变化。微分可以求出梯度函数，它给出曲线上任意一点的梯度。
The gradient function is often denoted by f'(x) or dy/dx. Once we know it, we can determine how steep the curve is and in which direction it is sloping at a specific x-value.
梯度函数通常记作 f'(x) 或 dy/dx。一旦我们得到它，就可以确定曲线在某个特定 x 值处的陡峭程度和倾斜方向。
3. Differentiation from First Principles: The Case of x² | 从第一原理推导 x² 的导数
The derivative of f(x) = x² can be found using the limit definition: f'(x) = lim (h → 0) [(x + h)² – x²] / h. Expanding (x + h)² gives x² + 2xh + h², so the numerator simplifies to 2xh + h².
函数 f(x) = x² 的导数可以通过极限定义求得：f'(x) = lim (h → 0) [(x + h)² – x²] / h。将 (x + h)² 展开得到 x² + 2xh + h²，因此分子简化为 2xh + h²。
f'(x) = lim (h → 0) (2xh + h²)/h = lim (h → 0) (2x + h) = 2x
Dividing by h leaves 2x + h, and letting h tend to 0 yields the result f' first-principles derivation justifies the short-cut rule we shall use for all polynomials.
除以 h 后得到 2x + h，令 h 趋于零就得出结果 f'(x) = 2x。这一从第一原理的推导为我们后面将用于所有多项式的快捷法则提供了依据。
4. The Power Rule for Differentiation | 幂函数求导法则
For any term of the form kxⁿ, where n is a real number, the derivative is nkxⁿ⁻¹. This is known as the power rule and is the fundamental tool for differentiating polynomials.
对于任何形如 kxⁿ 的项（n 为实数），其导数为 nkxⁿ⁻¹。这就是幂函数求导法则，是对多项式求导的基本工具。
If f(x) = kxⁿ, then f'(x) = nkxⁿ⁻¹
ower and then reduce the power by one. Constant terms (where n = 0) differentiate to zero.
我们将系数乘以原来的指数，再将指数减 1。常数项（n = 0）求导后得到零。
5. Differentiating y = x² | 求导 y = x²
tells us that the gradient at any point on the curve y = x² equals twice the x-coordinate.
对 y = x² 运用幂法则，系数为 1，指数为 2，则 dy/dx = 2×1×x²⁻¹ = 2x。这说明在曲线 y = x² 上任意一点的梯度都等于其 x 坐标的两倍。
f'(x) = 2x
The gradient is negative for x < 0, zero at the vertex (0,0) and positive for x > 0.
例如，在 x = 3 处梯度为 6；在 x = −2 处梯度为 −4。当 x < 0 时梯度为负，在顶点 (0,0) 处为零，当 x > 0 时梯度为正。
6. Differentiating the General Quadratic ax² + bx + c | 一般二次函数的求导
apply the power rule term by term. The derivative of ax² is 2ax, the derivative of bx is b, and the derivative of the constant c is 0.
要对 f(x) = ax² + bx + c 求导，我们逐项应用幂法则。ax² 的导数为 2ax，bx 的导数为 b，常数 c 的导数为 0。
f'(x) = 2ax + b
The resulting gradient function is a linear expression. This means that for any quadratic, the rate of change of y with respect to x is a straight-line function.
得到的梯度函数是一个线性表达式。这意味着对于任何二次函数，y 关于 x 的变化率都是一个直线函数。
7. Calculating Derivatives at Specific Points | 求特定点处的导数值
To find the gradient at a particular point, first find f' rentiating, then substitute the x-coordinate. For instance, if f(x) = 3x² − 2x + 1, then f'(x) = 6x − 2.
要求某一点处的梯度，先通过求导得到 f'(x)，再代入 x 坐标。例如，若 f(x) = 3x² − 2x + 1，则 f'(x) = 6x − 2。
At x = 1: f'(1) = 6(1) − 2 = 4
At x = −1, f' ative quickly is essential for problems involving tangents, normals and stationary points.
在 x = −1 处，f'(−1) = −8。快速计算导数值对于解决切线、法线和驻点问题至关重要。
8. Finding the Equation of a Tangent | 求切线方程
A tangent to a curve at point P(x₁, y₁) is a straight line that touches the curve and has the same gradient as the curve at that point. Its gradient m is f'(x₁).
曲线在点 P(x₁, y₁) 处的切线是一条与曲线相切且在该点处与曲线具有相同梯度的直线。其斜率 m 为 f'(x₁)。
y − y₁ = m(x − x₁)
First compute y₁ = f(x₁) and m = f' lope form. For a quadratic, the tangent equation can be simplified to the form y = mx + c.
首先计算 y₁ = f(x₁) 和 m = f'(x₁)，然后代入点斜式。对于二次函数，切线方程可以整理成 y = mx + c 的形式。
Example: For f(x) = x² − 4x + 3 at x = 2: y₁ = f(2) = −1, f'(x) = 2x − 4 ⇒ m = 0. Tangent: y − (−1) = 0(x − 2) ⇒ y = −1.
例子：对于 f(x) = x² − 4x + 3，在 x = 2 处：y₁ = f(2) = −1，f'(x) = 2x − 4 ⇒ m = 0。切线：y − (−1) = 0(x − 2) ⇒ y = −1。
9. Finding the Equation of a Normal | 求法线方程
The normal to a curve at a point is perpendicular to the tangent at that point. Its gradient mnormal is the negative reciprocal of the tangent gradient, provided the tangent gradient is not zero.
曲线在某一点的法线是垂直于该点切线的直线。其斜率 mnormal 为切线斜率的负倒数（前提是切线斜率不为零）。
mnormal = −1 / f'(x₁), if f'(x₁) ≠ 0
Using the same point (x₁, y₁), the normal equation is y − y₁ = mnormal (x − x₁). For the quadratic example above, at x = 1 with f'(1) = 4, the normal gradient is −1/4.
使用同一点 (x₁, y₁)，法线方程为 y − y₁ = mnormal (x − x₁)。对于上面二次函数的例子，在 x = 1 处 f'(1) = 4，则法线斜率为 −1/4。
10. Stationary Points of Quadratic Functions | 二次函数的驻点
A stationary point occurs where the gradient function is zero, i.e. f'(x) = 0. For a quadratic, setting f'(x) = 2ax + b = 0 gives the x-coordinate of the stationary point.
驻点出现在梯度函数为零的地方，即 f'(x) = 0。对于二次函数，令 f'(x) = 2ax + b = 0 即可得到驻点的 x 坐标。
x = −b / (2a)
Substituting this value back into the original quadratic gives the y-coordinate. The stationary point is the vertex (−b/(2a), f(−b/(2a))), which is the only turning point of a parabola.
将此值代回原二次函数就得到 y 坐标。驻点即为顶点 (−b/(2a), f(−b/(2a)))，这是抛物线唯一的转折点。
11. Using the Second Derivative | 使用二阶导数判断驻点性质
ative of a function is obtained by differentiating the first derivative. For a quadratic f'(x) = 2ax + b, differentiating again yields f''(x) = 2a, a constant.
二阶导数是对一阶导数再次求导得到。对于二次函数 f'(x) = 2ax + b，再次求导得 f''(x) = 2a，是一个常数。
f''(x) = 2a
If a > 0, then f''(x) > 0 for all x, meaning the stationary point is a minimum. If a < 0, then f''(x) < 0, so
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