Chapter 23

Find the derivative of each function. Check some by calculator. $$y=\frac{5}{\left(x^{2}-1\right)^{2}}$$

Evaluate each limit. $$\lim _{x \rightarrow 4} \frac{x^{2}-16}{x-4}$$

Find, by hand, all of the terms in each probability distribuisn, and graph. $$\text { If } y=3 x^{3}+2 x^{2}, \text { find } y^{\prime \prime}(2)$$

Find the derivative \(d x / d y\) of \(x\) with respect to \(y.\) $$x=(y-2)(y+3)^{5}$$

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Power Function with Negative Exponent. $$y=x^{-5}$$

Find the derivative of each function. Check some by calculator. $$y=\left(a-\frac{b}{x}\right)^{2}$$

Evaluate each limit. $$\lim _{x \rightarrow-3} \frac{x^{2}+2 x-3}{x+3}$$

Find, by hand, all of the terms in each probability distribuisn, and graph. $$\text { If } f(x)=x^{2}-4 x^{4}, \text { find } f^{\prime \prime}(3)$$

Find the derivative \(d x / d y\) of \(x\) with respect to \(y.\) $$x=\frac{y^{2}}{(4-y)^{3}}$$

Some of these can be multiplied out. For a few of these, take the derivative both before and after multiplying out, and compare the two. $$y=x^{2} \sqrt{3-4 x}$$

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Power Function with Negative Exponent. $$y=2 x^{-3}$$

Find the derivative of each function. Check some by calculator. $$y=\left(a+\frac{b}{x}\right)^{3}$$

Evaluate each limit. $$\lim _{x \rightarrow 1} \frac{x^{3}-x^{2}+2 x-2}{x-1}$$

The velocity of a moving point is given by the first derivative of the displacement, and the acceleration is given by the second derivative of the displacement. Find the velocity and acceleration at \(t=1.55 \mathrm{s}\), of a point whose displacement is given by $$s=4.55 t^{3}+2.85 t^{2}+5.22 \quad \mathrm{cm}$$ where \(t\) is the elapsed time, in seconds.

Find \(d y / d x\). (Treat \(a\) and \(r\) as constants.) $$5 x-2 y=7$$

Some of these can be multiplied out. For a few of these, take the derivative both before and after multiplying out, and compare the two. $$y=\sqrt{x}\left(3 x^{2}+2 x-3\right)$$

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Power Function with Negative Exponent. $$y=\frac{1}{x}$$

Limits Involving Zero or Infinity $$\lim _{x \rightarrow 0}\left(4 x^{2}-5 x-8\right)$$

Find the slope of the tangent or the rate of change at the given value of \(x\) $$y=\frac{1}{x^{2}} \text { at } x=1$$

Find the derivative of each function. Check some by calculator. $$y=\sqrt{1-3 x^{2}}$$

Find \(d y / d x\). (Treat \(a\) and \(r\) as constants.) $$2 x+3 y^{2}=4$$

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Power Function with Negative Exponent. $$y=\frac{1}{x^{2}}$$

Limits Involving Zero or Infinity $$\lim _{x \rightarrow 0} \frac{3-2 x}{x+4}$$

Find the slope of the tangent or the rate of change at the given value of \(x\) $$y=\frac{1}{x+1} \text { at } x=2$$

Find the derivative of each function. Check some by calculator. $$y=\sqrt{2 x^{2}-7 x}$$

Find \(d y / d x\). (Treat \(a\) and \(r\) as constants.) $$x y=5$$

Some of these can be multiplied out. For a few of these, take the derivative both before and after multiplying out, and compare the two. $$y=\left(2 x^{2}-3\right) \sqrt[3]{3 x+5}$$

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Power Function with Negative Exponent. $$y=\frac{3}{x^{3}}$$

Find the derivative of each function. Check some by calculator. $$y=\sqrt{1-2 x}$$

Find the slope of the tangent or the rate of change at the given value of \(x\) $$y=x+\frac{1}{x} \text { at } x=2$$

Find \(d y / d x\). (Treat \(a\) and \(r\) as constants.) $$x^{2}+3 x y=2 y$$

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Power Function with Negative Exponent. $$y=\frac{3}{2 x^{2}}$$

Limits Involving Zero or Infinity $$\lim _{x \rightarrow 0} \frac{3+x-x^{2}}{(x+3)(5-x)}$$

Find the derivative of each function. Check some by calculator. $$y=\frac{b}{a} \sqrt{a^{2}-x^{2}}$$

Find the slope of the tangent or the rate of change at the given value of \(x\) $$y=\frac{1}{x} \text { at } x=3$$

Find \(d y / d x\). (Treat \(a\) and \(r\) as constants.) $$y^{2}=4 a x$$

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Power Function with Fractional Exponent. $$y=7.5 x^{1 / 3}$$

Limits Involving Zero or Infinity $$\lim _{x \rightarrow 0}\left(\frac{1}{2+x}-\frac{1}{2}\right) \cdot \frac{1}{x}$$

Find the derivative of each function. Check some by calculator. $$y=\sqrt[3]{4-9 x}$$

Find the slope of the tangent or the rate of change at the given value of \(x\) $$y=2 x-3 \text { at } x=3$$

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Power Function with Fractional Exponent. $$y=4 x^{5 / 3}$$

Limits Involving Zero or Infinity $$\lim _{x \rightarrow 0} x \cos x$$

Find the derivative of each function. Check some by calculator. $$y=\sqrt[3]{a^{3}-x^{3}}$$

Find the slope of the tangent or the rate of change at the given value of \(x\) $$y=16 x^{2} \quad \text { at } x=1$$

Find \(d y / d x\). (Treat \(a\) and \(r\) as constants.) $$x^{3}+y^{3}-3 a x y=0$$

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Power Function with Fractional Exponent. $$y=4 \sqrt{x}$$

Limits Involving Zero or Infinity $$\lim _{x \rightarrow \infty} \frac{2 x+5}{x-4}$$

Find the derivative of each function. Check some by calculator. $$y=\frac{1}{\sqrt{x+1}}$$

Find the slope of the tangent or the rate of change at the given value of \(x\) $$y=2 x^{2}-6 \quad \text { at } x=3$$

Find \(d y / d x\). (Treat \(a\) and \(r\) as constants.) $$x^{2}+y^{2}=r^{2}$$