Inverse derivatives at corresponding points
| English | 中文 | Pinyin |
|---|---|---|
| inverse derivative | 反函数导数 | fǎn hán shù dǎo shù |
A calibration converts length to a reading. How do we reverse its local rate without taking the reciprocal of its output?
- A calibration converts length to a reading. How do we reverse its local rate without taking the reciprocal of its output?
- This lesson studies inverse derivative 反函数导数: The rate of the inverse output with respect to its input, evaluated at the corresponding original point.
Choose the mathematical structure
- Let f be one-to-one on a stated interval and g=f⁻¹. The identity f(g(v))=v gives f′(g(v))g′(v)=1. When the derivatives exist and f′(g(v))≠0, g′(v)=1/f′(g(v)). The inverse is evaluated at the original output v, not automatically at the same original input x. The reciprocal 1/f(x) is a different function.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines inverse derivative?
The rate of the inverse output with respect to its input, evaluated at the corresponding original point.
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 f(x)=x²+1 on x≥0, g(v)=√(v−1), v≥1. At v=10, g(10)=3 and f′(3)=6, so g′(10)=1/6. On x≤0 the inverse branch is −√(v−1), with g(10)=−3 and g′(10)=−1/6. For f=e^x, g=ln v: f′(g(v))=e^(lnv)=v, hence g′=1/v for v>0. For f=sin x on −π/2≤x≤π/2, g=arcsin v. Within −1<v<1, cos(g(v))=√(1−v²)>0, so g′=1/√(1−v²). The nonnegative cosine follows from the restricted sine branch; it is not an arbitrary sign choice.
Inverse derivatives at corresponding points
Let f be one-to-one on a stated interval and g=f⁻¹
Identify the appropriate changing variable and validate the derivative denominator or local branch.
For the positive quadratic branch, find g′(10).
The corresponding positive input is 3 and f′(3)=6, giving 1/6.
Test a tempting shortcut
- Do not replace the corresponding-point denominator f′(g(v)) by f′(v). One-to-one behaviour alone does not ensure a finite inverse gradient: x³ is invertible, but its derivative is zero at zero and cube root has no finite derivative there. Root/arcsine inverse endpoints can also lack finite gradients. Different inverse branches can give different signs.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The derivative of an inverse is always 1/f′(v), evaluated at the inverse input v. This claim is false. Explain which definition or assumption it violates.
For g(v)=ln v, find g′(4).
The exponential derivative at ln4 is 4, so its inverse rate is 1/4.
The derivative of an inverse is always 1/f′(v), evaluated at the inverse input v.
Do not replace the corresponding-point denominator f′(g(v)) by f′(v). One-to-one behaviour alone does not ensure a finite inverse gradient: x³ is invertible, but its derivative is zero at zero and cube root has no finite derivative there. Root/arcsine inverse endpoints can also lack finite gradients. Different inverse branches can give different signs.
Interpret a new situation
- Restrict the original domain, find the inverse input/output pair, differentiate f and evaluate its derivative at the recovered original input. Then take the reciprocal only if that finite gradient is nonzero. Swap horizontal and vertical units for the inverse rate. Check the result by differentiating an explicit inverse when one is available.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For the negative quadratic branch, find g′(10).
The corresponding negative input is −3 and f′(−3)=−6, giving −1/6.
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 rate of the inverse output with respect to its input, evaluated at the corresponding original point. Choose the relationship, show the method, check its assumptions and interpret the result.