Chain rules for nested functions and restricted inputs
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| composite derivative/ˈkɒmpəzɪt dɪˈrɪvətɪv/ | đạo hàm của hàm hợp |
How do we keep every link in its rate of change?
- A sensor converts a changing input through several formulas.
- 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.
The method
- 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.
For z=e^(x²), why is z′(0)=0 even though the exponential derivative is never zero?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
z′=e^(x²)·2x. The outer factor e^(x²) is always positive, but the inner rate 2x vanishes at 0, so the product is zero - a flat point on the curve, not a rule failure.
A worked case
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.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
4(−1)³×3=−12.
Where it goes wrong
- 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.
Not true: The derivative of e^(x²) is e^(x²) without an inner-rate factor. The chain rule multiplies by the inner derivative: the derivative of e^(x²) is 2x·e^(x²).
For e^(x²), find the derivative at x=0.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The inner rate is 2x, which is zero at zero.
The chain rule multiplies by the inner derivative.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. The chain rule multiplies by the inner derivative: the derivative of e^(x²) is 2x·e^(x²).
Applying it
- 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.
For ln(2x+1), find the derivative at x=1. Give it to 3 significant figures.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The inner rate is 2; 2/(2×1+1)=2/3.
Match each composite to its derivative.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Differentiate the outer function AT the inner expression, then multiply by the inner derivative; a nested chain multiplies every link.
In your course
- AQA A-level Mathematics (7357), specification section G.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Composite derivative: The outer derivative evaluated at the inner output, multiplied by the derivative of that inner input.
- For y=f(u) and u=g(x), dy/dx=f′(g(x))g′(x).
- Watch for: do not stop after differentiating the outer power or exponential.