Chain rules for nested functions and restricted inputs
| English | 中文 | Pinyin |
|---|---|---|
| composite derivative | 复合函数导数 | fù hé hán shù dǎo shù |
A sensor converts a changing input through several formulas. How do we keep every link in its rate of change?
- A sensor converts a changing input through several formulas. How do we keep every link in its rate of change?
- This lesson studies composite derivative 复合函数导数: The outer derivative evaluated at the inner output, multiplied by the derivative of that inner input.
Choose the mathematical structure
- For y=f(u) and u=g(x), dy/dx=f′(g(x))g′(x). Evaluate the outer derivative at the actual inner expression. For several nested functions, multiply every inner-rate factor. Domain restrictions come from each stage; the differentiated expression cannot restore a forbidden original input.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines composite derivative?
The outer derivative evaluated at the inner output, multiplied by the derivative of that inner input.
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, set u=3x−1: y′=4u³×3=12(3x−1)³, giving y′(0)=−12. For z=e^(x²), z′=e^(x²)×2x; z′(0)=0 although the outer exponential derivative is never zero. For w=ln(2x+1), x>−1/2, w′=2/(2x+1); w′(1)=2/3. For r=sin((2x+1)²), first set u=(2x+1)²: r′=cos((2x+1)²)×2(2x+1)×2=4(2x+1)cos((2x+1)²). For s=√(5−x²), the original domain is −√5≤x≤√5, but s′=−x/√(5−x²) is finite only in the open interval.
Chain rules for nested functions and restricted inputs
For y=f(u) and u=g(x), dy/dx=f′(g(x))g′(x)
Choose the derivative rule and preserve every coefficient, inner rate and input restriction.
For (3x−1)^4, find the derivative at x=0.
4(−1)³×3=−12.
Test a tempting shortcut
- Do not stop after differentiating the outer power or exponential. For sin(u²), the cosine input stays u²; it is not replaced by 2u. Missing one link in a nested derivative loses a factor. A zero inner derivative can give zero total derivative, so dividing by it without checking can lose valid cases. A square root can exist at an endpoint without a finite derivative there.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The derivative of e^(x²) is e^(x²) without an inner-rate factor. This claim is false. Explain which definition or assumption it violates.
For e^(x²), find the derivative at x=0.
The inner rate is 2x, which is zero at zero.
The derivative of e^(x²) is e^(x²) without an inner-rate factor.
Do not stop after differentiating the outer power or exponential. For sin(u²), the cosine input stays u²; it is not replaced by 2u. Missing one link in a nested derivative loses a factor. A zero inner derivative can give zero total derivative, so dividing by it without checking can lose valid cases. A square root can exist at an endpoint without a finite derivative there.
Interpret a new situation
- Work from the outside inward and record each input change. Multiply the factors, then simplify. For a short polynomial, expansion gives an independent alternative check. Retain every log/root restriction and interpret whether a zero rate comes from the outer function, the inner input or both.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For ln(2x+1), find the derivative at x=1.
The inner rate is 2; 2/(2×1+1)=2/3.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · G. Match the target tier and specification before assigning extensions.
- 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 outer derivative evaluated at the inner output, multiplied by the derivative of that inner input. Choose the relationship, show the method, check its assumptions and interpret the result.