Find the derivative of the following function by first expanding the expression. \[ f(x)=(6 x+1)\left(5 x^{2}+7\right) \] \( f^{\prime}(x)=\square \)

Explain how accumulation functions can be interpreted in terms of areas under a curve.

DERIVADAS DE PRODUCTOS Y COCIENTES 1. \( h(z)=(2-\sqrt{z})\left(3+8 \sqrt[3]{z^{2}}\right) \) 2. \( f(x)=\left(x-\frac{2}{x}\right)\left(7-2 x^{3}\right) \) 3. \( y=\left(x^{2}-5 x+1\right)\left(12+2 x-x^{3}\right) \) 4. \( g(x)=\frac{\sqrt[3]{x}}{1+x^{2}} \) 5. \( Z(y)=\frac{4 y-y^{2}}{6-y} \) 6. \( V(t)=\frac{1-10 t+t^{2}}{5 t+2 t^{3}} \) 7. \( f(w)=\frac{(1-4 w)(2+w)}{3+9 w} \)

Problemas: En cada caso indique cual es el factor de integración de la ecuación diferencial lineal de primer orden y encuentre la solución general. \( \begin{array}{ll}\text { i. }\left(x+\frac{y}{x}\right) d x-d y=0 & \text { iii. } x y^{\prime}+4 y=9 x^{5}+2 x^{3} \\ \text { ii. } y^{\prime}-\frac{y}{x}=x^{4} & \text { iv. } x y^{\prime}+2 y=3 x \\ \text { v. } y^{\prime}-2 y=3 e^{2 x} & \end{array} \)

5- Qual das seguintes funções é solução da equação diferencial \( y^{\prime \prime}+2 y^{\prime}-3 y=0 \) :

\( H(x)=\frac{3}{x^{2}+1} \) dos funciones \( f(x) \) y \( g(x) \) tales que \( (f \circ g)(x)=H(x) \) son: