In applications, derivative problems usually are hard not because Differential Calculus is hard but because most of the problems require additional mathematical problems to solve which are mostly algebra and transcendental functions.
In one of the examples being demonstrated and discussed by the professor for finding the area enclosed by the slope and the y and x axis from the curve \( \frac{1}{x} \), after solving for the x-intercept with substituting the values on the slope equation, he didn’t do the same for solving the y-intercept but instead he used symmetry so that inferring from the previous solution to the x-intercept
\[ x=2x_0 \]he can quickly solve for the y-intercept which is
\[ y=2y_0 \]and such stems from the fact of the symmetry that \( x = \frac{1}{y} \implies xy = 1 \implies y = \frac{1}{x} \).
Symmetry here is only true regardless of the shape of triangles on both sides as long as the area on both sides are equal.
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