91Ó°ÊÓ

Find \(d y / d x\) \(y=\frac{1+\csc \left(x^{2}\right)}{1-\cot \left(x^{2}\right)}\)

Find \(d y /\left.d x\right|_{x=-2}\), given that \(y=(x+2) / x\).

Find an equation for the line that is tangent to the curve \(y=x^{3}-2 x+1\) at the point \((0,1)\), and use a graphing utility to graph the curve and its tangent line on the same screen.

Find \(d y / d x\) \(y=(5 x+8)^{7}(1-\sqrt{x})^{6}\)

Determine whether the statement is true or false. Explain your answer. $$ \begin{aligned} &\text { If } f^{\prime}(2)=5 \text { , then } \\ &\left.\frac{d}{d x}\left[4 f(x)+x^{3}\right]\right|_{x=2}=\left.\frac{d}{d x}[4 f(x)+8]\right|_{x=2}=4 f^{\prime}(2)=20 \end{aligned} $$

Find all values of \(x\) at which the tangent line to the given curve satisfies the stated property. $$ y=\frac{1}{x+4} ; \text { passes through the origin } $$

Find \(d y / d x\) \(y=\left(x^{2}+x\right)^{5} \sin ^{8} x\)

Use a graphing utility to graph the following on the same screen: the curve \(y=x^{2} / 4\), the tangent line to this curve at \(x=1\), and the secant line joining the points \((0,0)\) and \((2,1)\) on this curve.

Determine whether the statement is true or false. Explain your answer. $$ \begin{aligned} &\text { If } f(x)=x^{2}\left(x^{4}-x\right), \text { then } \\ &\qquad f^{\prime \prime}(x)=\frac{d}{d x}\left[x^{2}\right] \cdot \frac{d}{d x}\left[x^{4}-x\right]=2 x\left(4 x^{3}-1\right) \end{aligned} $$

Let \(f(x)=2^{x}\). Estimate \(f^{\prime}(1)\) by (a) using a graphing utility to zoom in at an appropriate point until the graph looks like a straight line, and then estimating the slope (b) using a calculating utility to estimate the limit in Formula (13) by making a table of values for a succession of values of \(w\) approaching \(1 .\)