Let \( x y+e^{x}-e^{y}=0 \), then \( y^{\prime}(0)= \)

A solid is formed with triangular cross-sections where each triangle's base is defined by the equation \(y = x^2\) on interval \([0, 2]\). What is the volume of this solid?

Choose the correct answer from those givens: (1) If \( \operatorname{Lim}_{x \rightarrow 0} \frac{\cos x-\cos ^{a} x}{x^{2}}=1 \), then \( a=\ldots \ldots \ldots \ldots \ldots \) \( \begin{array}{llll}\text { (a) } 1 & \text { (b) zero } & \text { (c) } 2 & \text { (d) } 3\end{array} \)

Calcula el área entre las curvas \( y = x^3 - 3x \) y \( y = x + 1 \) sobre el intervalo definido por sus intersecciones.

Application 3: Simplifier les expressions suivantes: \( \begin{array}{ll}A=\ln (\sqrt{3})+\ln 3-\ln 9 & ; \\ C=\ln \left(\frac{1}{3}\right)+\ln \left(\frac{3}{5}\right)+\ln \left(\frac{5}{7}\right)+\ln \left(\frac{7}{9}\right) & ;\end{array} \quad D=\ln (\sqrt[3]{2}-\ln 16 \)

Find the derivative of the function. \[ y=\frac{7}{x^{4}}-\frac{2}{x} \]