Chain, product, quotient and implicit differentiation · 链式法则、乘积法则、商法则和隐函数求导
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| chain rule/tʃeɪn ruːl/ | 链式法则 | liàn shì fǎ zé |
Does the inside expression change too?
- A cost curve is a power of a changing expression. Differentiating the outer power alone misses the rate of its input.
- This lesson studies chain rule 链式法则: The rule that multiplies the outer derivative by the inner derivative for a composite function.
Choose the mathematical structure
- For y=f(g(x)), y prime=f prime(g(x))g prime(x). For uv, differentiate to u prime v+uv prime. For u/v, use (u prime v-uv prime)/v², where v≠0.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines chain rule? · 以下哪项描述正确定义了链式法则？
The rule that multiplies the outer derivative by the inner derivative for a composite function. · 对于复合函数，将外层函数的导数乘以内层函数的导数的规则。
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For y=(3x+1)^4, y prime=4(3x+1)^3×3=12(3x+1)^3. At x=0, the gradient is 12. For x²+y²=25, differentiate implicitly: 2x+2y y prime=0, so y prime=-x/y when y≠0.
Chain, product, quotient and implicit differentiation · 链式法则、乘积法则、商法则和隐函数求导
For y=f(g(x)), y prime=f prime(g(x))g prime(x) · 对于 y=f(g(x))，y'=f'(g(x))g'(x)
Compare the model with the worked case and explain one change. · 对比模型与已解案例，并说明其中一处变化。
Find the derivative of (3x+1)^4 at x=0. · 求 (3x+1)^4 在 x=0 处的导数。
Chain rule gives 4(3x+1)³×3; at zero this is 12. · 链式法则给出 4(3x+1)³×3；在零处该值为 12。
Test a tempting shortcut
- A derivative of a product is not the product of the derivatives. In implicit differentiation, every differentiated function of y brings a dy/dx factor. A quotient's denominator is squared.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The derivative of u(x)v(x) is always u prime times v prime. This claim is false. Explain which definition or assumption it violates.
For x²+y²=25, find dy/dx at (3,4). · 对于 x²+y²=25，求在 (3,4) 处的 dy/dx。
Implicit differentiation gives 2x+2y y prime=0, so y prime=-3/4. · 隐函数求导得 2x+2y·y'=0，因此 y'=-3/4。
The derivative of u(x)v(x) is always u prime times v prime. · u(x)v(x) 的导数总是 u' 乘以 v'。
A derivative of a product is not the product of the derivatives. In implicit differentiation, every differentiated function of y brings a dy/dx factor. A quotient's denominator is squared. · 乘积的导数不等于导数的乘积。在隐函数求导中，每个对 y 的函数求导都会引入一个 dy/dx 因子。商法则的分母要平方。
Interpret a new situation
- Choose a useful form before differentiating: expanding a short polynomial may be simpler. For related rates, write the relation in symbols, differentiate with respect to time, then substitute measured values.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For y=xe^x, find dy/dx at x=0. · 对于 y=xe^x，求在 x=0 处的 dy/dx。
Product rule gives e^x+xe^x; at zero this is 1. · 乘积法则得 e^x+xe^x；在零处该值为 1。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- edexcel IAL pure mathematics; official unit P4. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The rule that multiplies the outer derivative by the inner derivative for a composite function. Choose the relationship, show the method, check its assumptions and interpret the result.