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Use df” ‘( (x) to find the derivative of the inverse at the indicated point dx F[(x)] Let flx) = Sx 1,x20. Find df-1, (x)l [Note that f(1) =4.] dx X=4 dr L ()l dx X=4 (Simplify your answer:
The Correct Answer and Explanation is:
To find the derivative of the inverse of a function f(x)f(x)f(x) at a specific point, we can use the formula: ddxf−1(x)=1f′(f−1(x))\frac{d}{dx} f^{-1}(x) = \frac{1}{f'(f^{-1}(x))}dxdf−1(x)=f′(f−1(x))1
Given that f(x)=x+1f(x) = \sqrt{x} + 1f(x)=x+1 and f(1)=4f(1) = 4f(1)=4, we are tasked with finding ddxf−1(x)\frac{d}{dx} f^{-1}(x)dxdf−1(x) at x=4x = 4x=4.
Step-by-step Solution:
- Given function: f(x)=x+1f(x) = \sqrt{x} + 1f(x)=x+1
- Find f−1(x)f^{-1}(x)f−1(x):
To find the inverse of f(x)f(x)f(x), we need to express xxx in terms of yyy (where y=f(x)y = f(x)y=f(x)): y=x+1y = \sqrt{x} + 1y=x+1 Solve for xxx in terms of yyy: y−1=xy – 1 = \sqrt{x}y−1=x Squaring both sides: (y−1)2=x(y – 1)^2 = x(y−1)2=x Therefore, the inverse function is: f−1(x)=(x−1)2f^{-1}(x) = (x – 1)^2f−1(x)=(x−1)2 - Find f′(x)f'(x)f′(x):
The derivative of f(x)=x+1f(x) = \sqrt{x} + 1f(x)=x+1 is: f′(x)=ddx(x+1)=12xf'(x) = \frac{d}{dx} (\sqrt{x} + 1) = \frac{1}{2\sqrt{x}}f′(x)=dxd(x+1)=2x1 - Find the derivative of the inverse function f−1(x)f^{-1}(x)f−1(x) at x=4x = 4x=4: We know from the formula: ddxf−1(x)=1f′(f−1(x))\frac{d}{dx} f^{-1}(x) = \frac{1}{f'(f^{-1}(x))}dxdf−1(x)=f′(f−1(x))1 First, we need to find f−1(4)f^{-1}(4)f−1(4): f−1(4)=(4−1)2=9f^{-1}(4) = (4 – 1)^2 = 9f−1(4)=(4−1)2=9 Now, evaluate f′(f−1(4))=f′(9)f'(f^{-1}(4)) = f'(9)f′(f−1(4))=f′(9): f′(9)=129=16f'(9) = \frac{1}{2\sqrt{9}} = \frac{1}{6}f′(9)=291=61
- Find ddxf−1(x)\frac{d}{dx} f^{-1}(x)dxdf−1(x) at x=4x = 4x=4: Using the formula: ddxf−1(4)=1f′(9)=116=6\frac{d}{dx} f^{-1}(4) = \frac{1}{f'(9)} = \frac{1}{\frac{1}{6}} = 6dxdf−1(4)=f′(9)1=611=6
Final Answer:
The derivative of the inverse of the function f(x)=x+1f(x) = \sqrt{x} + 1f(x)=x+1 at x=4x = 4x=4 is: 6\boxed{6}6
Explanation:
To compute the derivative of the inverse function, we used the relationship between a function and its inverse. First, we found the inverse of the given function and then used the formula for the derivative of the inverse. The derivative of the inverse is the reciprocal of the derivative of the function at the corresponding value. Thus, we evaluated f′(x)f'(x)f′(x) at x=f−1(4)x = f^{-1}(4)x=f−1(4) and found the final answer of 6.
