\[\eqalign{\frac{dy}{dx} &=& \frac{d(4x - x^3)}{dx}\\ &=& 4-3x^2\\}\]

\[\eqalign{ \frac{dy}{dx} &=& lim_{d\rightarrow 0} \left(\frac {4(x+d) - (x+d)^3 - (4x - x^3)}{d}\right)\\ &=&lim_{d\rightarrow 0} \left(\frac{4x + 4d -x^3 -3dx^2 - 3d^2x - d^3 - 4x + x^3}{d}\right)\\ &=&lim_{d\rightarrow 0} \left(\frac{ 4d -3dx^2 - 3d^2x - d^3}{d}\right)\\ &=&lim_{d\rightarrow 0} \left(4 -3x^2 - 3dx - d^2 \right) = 4 -3x^2 \\ }\]

\[\eqalign{ area_{rthnd} &=& h \left(f(x_1) + f(x_2) + ... + f(x_4)\right)\\ area_{midpt} &=& h \left({\Large f}\left(\frac{x_0 +x_1}{2}\right) + {\Large f}\left(\frac{x_1 +x_2}{2}\right) + ...+{\Large f}\left(\frac{x_3 +x_4}{2}\right)\right)\\ area_{trap} &=& \frac{h}{2}\left(f\left(x_0\right) + 2 f\left(x_1\right) + 2 f\left(x_2\right)+ + 2 f\left(x_3\right) + + 2 f\left(x_1\right)\right)\\ area_{simp}&=& \frac{h}{3}\left(f\left(x_0\right) + 4 f\left(x_1\right) + 2 f\left(x_2\right) + 4 f\left(x_3\right) + f\left(x_4\right)\right)\\ area_{int} &=& \int_{x_0}^{x_4} f(x)dx\\ }\]