The derivative · La derivada
| English | Español |
|---|---|
| derivative/dɪˈrɪvətɪv/ | derivada |
| gradient of the tangent/ˈɡreɪdɪənt ɒvðə ˈtændʒənt/ | pendiente de la tangente |
| differentiation from first principles/ˌdɪfəˌrenʃɪˈeɪʃn frɒm fɜːst ˈprɪnsɪplz/ | diferenciación por definición |
| power rule/ˈpaʊə ruːl/ | regla de la potencia |
| product rule/ˈprɒdʌkt ruːl/ | regla del producto |
| quotient rule/ˈkwəʊʃənt ruːl/ | regla del cociente |
| chain rule/tʃeɪn ruːl/ | regla de la cadena |
| stationary point/ˈsteɪʃənəri pɔɪnt/ | punto estacionario |
| second derivative/ˈsekənd dɪˈrɪvətɪv/ | segunda derivada |
| maximum/ˈmæksɪməm/ | máximo |
| minimum/ˈmɪnɪməm/ | mínima |
A finite average hides local change
- For · A favor $y=x^2$, 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 · Desliza el segundo punto hacia el primero
$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
The chord becomes the tangent, and its gradient becomes the derivative. · La cuerda se convierte en la tangente, y su pendiente se convierte en la derivada.
What does the derivative of a function at a point measure? · ¿Qué mide la derivada de una función en un punto?
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. · Cuando existe, la derivada es una tasa de cambio límite instantánea y la pendiente tangente. Una integral definida mide acumulación firmada, que no necesariamente equivale al área geométrica.
Take a limit without dividing by zero
- Differentiation from first principles 从定义求导 uses · usa $f'(x)=\lim_{h\to0}[f(x+h)-f(x)]/h$. 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 · A favor $f(x)=x^2$, 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 · El 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 · El 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. · Diferencia y = x⁵. Da la derivada.
Bring the power down and reduce it by one: 5x⁴. · Baja el exponente y redúcelo en uno: 5x⁴.
Which rule differentiates y = (3x + 1)⁴? · ¿Qué regla diferencia y = (3x + 1)⁴?
A function inside a function. The derivative is 4(3x + 1)³ × 3 = 12(3x + 1)³. · Una función dentro de otra función. La derivada es 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? · y = x³ − 3x tiene un punto estacionario en un valor positivo de x. ¿Cuál es?
dy/dx = 3x² − 3 = 0 gives x = ±1, so the positive one is x = 1. · dy/dx = 3x² − 3 = 0 da x = ±1, por lo tanto el positivo es x = 1.
At a stationary point the second derivative is negative. What kind of point is it? · En un punto estacionario, la segunda derivada es negativa. ¿Qué tipo de punto es?
Negative second derivative means the gradient is falling through zero, so the curve turns downward. · Segunda derivada negativa significa que la pendiente está descendiendo a través de cero, por lo que la curva se vuelve cóncava hacia abajo.
Put the steps of a stationary-point question in the order that earns the marks. · Coloca los pasos de una pregunta sobre puntos estacionarios en el orden que otorga los puntos.
Find candidate x-values from the derivative, substitute into the original function for their coordinates, and justify classification. A zero second derivative is inconclusive. · Encuentre valores candidatos de x a partir de la derivada, sustitúyalos en la función original para sus coordenadas, y justifique la clasificación. Una segunda derivada cero es inconclusiva.
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.