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Why is the inverse of a gradient -1/x?

How can I determine the characteristics of a curve on an x-y set of axis (eg. points of intersection, stationary points, area under graph)?

Let f(x)= x^3 -9x^2 -81x + 12. Calculate f'(x) and f''(x). Use f'(x) to calculate the x-values of the stationary points of this function.

Integrate, by parts, y=xln(x),

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