Implicit differentiation and inverse trigonometric functions practice question 15
The function is one-to-one. Find .
You do not need a formula for . How are the slopes of a function and its inverse related at matching points?
If , then .
Find by inspection: which small integer has ?
A number, as a fraction or a decimal.
Theory UBC Math 100 §4.6CLP-1 §2.12Key results
Worked solution
Strategy
The derivative of an inverse at a point is the reciprocal of the derivative of the function at the corresponding point. Find that point first, then the reciprocal slope.
Working
The point on f that maps to 4. Uniqueness follows from f being one-to-one.
Slope of f at that point.
Reflecting a graph in y = x takes slope m to slope 1/m.
Answer
The answer is one fifth.
Check
f(1.01) ≈ 4.0503, so f−¹ moves from 1 to 1.01 as its input moves from 4 to 4.0503, a slope of about 0.2.
Where this goes wrong
Computing 1/f′(4) = 1/50. The derivative of f must be taken at f−¹(4) = 1, the point on the original curve, not at 4.
The idea behind it
The implicit differentiation and inverse trigonometric functions lesson, a figure to move, in a few steps
Leans on
One-to-one functions and inverses, from Functions, graphs and inverses
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