Question - Calculating the Inverse of a Derivative at a Specific Point
Solution:
To solve for the value of $$(df/dx)^{-1}$$ at the point $$x = 398 = f(5)$$, we are essentially looking to compute the reciprocal of the derivative of $$f(x)$$ at $$x = 5$$. The reciprocal of the derivative is also known as the derivative of the inverse function evaluated at the corresponding y-value, which in this case is 398.First, we'll find $$df/dx$$ by differentiating $$f(x)$$:$$f(x) = 5x^3 - 9x^2 - 2$$$$df/dx = 15x^2 - 18x$$Now we need to evaluate $$df/dx$$ at $$x = 5$$:$$df/dx|_{x=5} = 15(5)^2 - 18(5) = 15(25) - 90 = 375 - 90 = 285$$Now, $$(df/dx)^{-1}$$ at $$x = 398$$ is the reciprocal of this value:$$(df/dx)^{-1}|_{f(5)} = 1 / df/dx|_{x=5} = 1/285$$So, the value is $$\frac{1}{285}$$, which is the simplified fraction we were looking to find.
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