\( f ( x ) = \left\{ \begin{array} { l l l } { \frac { 1 } { x - 4 } } & { n p u } & { x < 4 } \\ { - x + 5 } & { n p u } & { 4 \leq x < 5 } \\ { \log _ { 5 } x } & { n p u } & { x \geq 5 } \end{array} \right. \)

Calcular el área de las siguientes regiones que están delimitadas por las siguientes curvas cuyas ecuaciones son: A) \( \mathrm{y}=\mathrm{x}+6 \quad ; \quad \mathrm{y}=\mathrm{x}^{3} \quad ; 2 \mathrm{y}+\mathrm{x}=0 \)

Calculate the derivative of the following function. \( y=\sqrt{17 x+4} \)

A. ctangle initially has dimensions 2 cm by 7 cm . All sides begin increasing in length at a rate of \( 2 \mathrm{~cm} / \mathrm{s} \). At what rate is the area of the rectangle increasing after 16 s ? Let \( \mathrm{A}, \mathrm{b} \), and h be the area, base, and height of a rectangle, respectively. Write an equation relating \( \mathrm{A}, \mathrm{b} \), and h . \( \mathrm{A}=\mathrm{b} \times \mathrm{h} \) Differentiate both sides of the equation with respect to t . \( \mathrm{dA}= \)

Find the derivative of the following function. \[ f(x)=\frac{x^{3}-9 x^{2}+x}{x-1} \]

Find the following limit or state that it does not exist. \( \lim _{h \rightarrow 0} \frac{\sqrt{225+h}-15}{h} \)