The derivative
Introduced| English | Meaning |
|---|---|
| derivative/dɪˈrɪvətɪv/ | The instantaneous rate of change of a function with respect to its input. |
| gradient of the tangent/ˈɡreɪdɪənt ɒvðə ˈtændʒənt/ | The slope of a tangent line to a curve at a point, equal to the derivative there when it exists. |
| differentiation from first principles/ˌdɪfəˌrenʃɪˈeɪʃn frɒm fɜːst ˈprɪnsɪplz/ | Finding a derivative from the limit of a difference quotient as the change in the independent variable approaches zero. |
| power rule/ˈpaʊə ruːl/ | The differentiation rule stating that the derivative of x raised to n is n multiplied by x raised to (n - 1). |
| product rule/ˈprɒdʌkt ruːl/ | The differentiation rule for two multiplied functions: the first times the derivative of the second plus the second times the derivative of the first. |
| quotient rule/ˈkwəʊʃənt ruːl/ | The differentiation rule for a ratio: the denominator times the numerator's derivative minus the numerator times the denominator's derivative, divided by the denominator squared. |
| chain rule/tʃeɪn ruːl/ | The rule for differentiating a composite function by multiplying the appropriate derivatives. |
| stationary point/ˈsteɪʃənəri pɔɪnt/ | A point on a differentiable curve where its derivative is zero. |
| second derivative/ˈsekənd dɪˈrɪvətɪv/ | The derivative of a function's first derivative, describing how its gradient changes. |
| maximum/ˈmæksɪməm/ | A value at least as large as all other values in the specified domain, or in a neighbourhood for a local maximum. |
| minimum/ˈmɪnɪməm/ | A value at least as small as all other values in the specified domain, or in a neighbourhood for a local minimum. |
A finite average hides local change
- For $y=x^2$, the secant from x equal to 1 to 2 has slope 3, but the tangent at x equal to 1 has slope 2.
- A derivative 导数 is an instantaneous limiting rate, when it exists. Geometrically it gives the gradient of the tangent 切线斜率 at a differentiable point.
Slide the second point towards the first
$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
The chord becomes the tangent, and its gradient becomes the derivative.
What does the derivative of a function at a point measure?
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Where it exists, the derivative is an instantaneous limiting rate of change and tangent gradient. A definite integral measures signed accumulation, which need not equal geometric area.
Take a limit without dividing by zero
- Differentiation from first principles 从定义求导 uses $f'(x)=\lim_{h\to0}[f(x+h)-f(x)]/h$. The difference quotient uses nonzero h before its limit is taken.
- One-sided limits must agree for a two-sided derivative. A sharp corner can prevent differentiability even if the function is continuous.
A power from its definition. For $f(x)=x^2$, the quotient is $[(x+h)^2-x^2]/h=2x+h$ for nonzero h. Its limit is $2x$. For $|x|$ at zero, the quotient $|h|/h$ instead has right-hand limit 1 and left-hand limit negative 1, so no derivative exists there.
Select a rule for the function form
- The power rule 幂法则 gives $(x^n)'=nx^{n-1}$ on a valid domain. The product rule 乘积法则 gives $(uv)'=u'v+uv'$.
- The quotient rule 商法则 gives $(u/v)'=(u'v-uv')/v^2$ where v is nonzero. The chain rule 链式法则 gives $[f(g(x))]'=f'(g(x))g'(x)$, including the inner derivative.
Differentiate y = x⁵. Give the derivative.
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Bring the power down and reduce it by one: 5x⁴.
Which rule differentiates y = (3x + 1)⁴?
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A function inside a function. The derivative is 4(3x + 1)³ × 3 = 12(3x + 1)³.
A stationary point needs classification
- A stationary point 驻点 has $f'(x)=0$. A nonzero second derivative 二阶导数 there gives a local maximum 极大值 when negative or minimum 极小值 when positive.
- A zero second derivative is inconclusive. Examine first-derivative signs or other evidence; $x^3$ has a stationary inflection at zero, not a maximum or minimum.
y = x³ − 3x has a stationary point at a positive value of x. What is it?
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dy/dx = 3x² − 3 = 0 gives x = ±1, so the positive one is x = 1.
At a stationary point the second derivative is negative. What kind of point is it?
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Negative second derivative means the gradient is falling through zero, so the curve turns downward.
Put the steps of a stationary-point question in the order that earns the marks.
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Find candidate x-values from the derivative, substitute into the original function for their coordinates, and justify classification. A zero second derivative is inconclusive.
Find both coordinates and describe intervals. In sheet 3.1, factor the derivative to find candidate x-values, substitute into the original function for y, then test classification and where the curve increases or decreases.
$dy/dx$ differentiates with respect to x; a dot over y normally differentiates with respect to time. State the independent variable rather than treating these notations as interchangeable in every problem. Trigonometric derivative rules here use radians.