Calculating the Inverse of a Derivative at a Specific Point
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.
