The table above gives selected values for a differentiable and decreasing function f and its derivative. If g(x)=f−1(x) for all x, which of the following is a correct expression for g′(2) ?
Correct. This value can be confirmed using the chain rule and the definition of an inverse function. Since f(g(x))=xf(g(x))=x, it follows that ddxf(g(x))=f′(g(x))g′(x)=ddx(x)=1⇒g′(x)=1f′(g(x))ddxf(g(x))=f′(g(x))g′(x)=ddx(x)=1⇒g′(x)=1f′(g(x)). Therefore, g′(2)=1f′(g(2))=1f′(4)=−15g′(2)=1f′(g(2))=1f′(4)=−15.
The derivative of an inverse function can be found using the formula:
(g⁻¹(x))’ = 1 / f'(g⁻¹(x))
We are given values for f and f’ in the table, but we need to find the value of g'(2).
To find g'(2), we need to find g⁻¹(2), which is the value of x such that f(x) = 2. From the table, we see that f(3) = 2, so g⁻¹(2) = 3.
Now we can use the formula for the derivative of the inverse function:
g'(2) = 1 / f'(g⁻¹(2)) = 1 / f'(3) = 1 / (-1) = -1
Therefore, the correct expression for g'(2) is -1.
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