Chapter 23

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. $$y=x\left(x^{2}-3\right)$$

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Derivative of a Constant. $$y=8$$

Find the derivative of each function. Check some by calculator. $$y=(2 x+1)^{5}$$

Graph the given function. Then find the slope or rate of change of the curve at the given value of \(x\), either manually, by zooming in, by using the TANGENT feature on your calculator, or numerically, as directed by your instructor. $$y=x^{2} \quad \text { at } x=2$$

Evaluate each limit. $$\lim _{x \rightarrow 2}\left(x^{2}+2 x-7\right)$$

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. $$y=x^{3}(5-2 x)$$

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Derivative of a Constant. $$y=\pi$$

Find the derivative of each function. Check some by calculator. $$y=\left(2-3 x^{2}\right)^{3}$$

Find the derivative by the delta method. $$y=4 x-3$$

Graph the given function. Then find the slope or rate of change of the curve at the given value of \(x\), either manually, by zooming in, by using the TANGENT feature on your calculator, or numerically, as directed by your instructor. $$y=-x^{2} \text{ at } x=3$$

Evaluate each limit. $$\lim _{x \rightarrow-1}\left(x^{3}-3 x^{2}-5 x-5\right)$$

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. $$y=x\left(x^{2}-2\right)^{2}$$

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Derivative of a Constant. $$y=a^{2}$$

Derivatives with Respect to Other Variables. If \(w=y^{2}+u^{3},\) find \(d w / d u.\)

Find the derivative of each function. Check some by calculator. $$y=\left(3 x^{2}+2\right)^{4}-2 x$$

Graph the given function. Then find the slope or rate of change of the curve at the given value of \(x\), either manually, by zooming in, by using the TANGENT feature on your calculator, or numerically, as directed by your instructor. $$y=\sqrt{x} \quad \text { at } x=1$$

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

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. $$y=x(x-9)^{3}$$

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Derivative of a Constant. $$y=3 b+7 c$$

Find the derivative of each function. Check some by calculator. $$y=\left(x^{3}+5 x^{2}+7\right)^{2}$$

Find the derivative by the delta method. $$y=7-4 x$$

Graph the given function. Then find the slope or rate of change of the curve at the given value of \(x\), either manually, by zooming in, by using the TANGENT feature on your calculator, or numerically, as directed by your instructor. $$y=\sqrt{x}+x^{2} \quad \text { at } x=2$$

Evaluate each limit. $$\lim _{x \rightarrow 5} \frac{5+4 x-x^{2}}{5-x}$$

Find, by hand, all of the terms in each probability distribuisn, and graph. $$y=3 x^{4}-x^{3}+5 x$$

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Derivative of a Constant Times a Power Function. $$y=x$$

Find the derivative of each function. Check some by calculator. $$y=(2-5 x)^{3 / 5}$$

Graph the given function. Then find the slope or rate of change of the curve at the given value of \(x\), either manually, by zooming in, by using the TANGENT feature on your calculator, or numerically, as directed by your instructor. $$y=x-x^{2} \quad \text{ at } x=3$$

Evaluate each limit. $$\lim _{x \rightarrow 5} \frac{x^{2}-25}{x-5}$$

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. $$y=(7-2 x)(x+4)$$

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Derivative of a Constant Times a Power Function. $$y=3 x$$

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

Find the derivative by the delta method. $$y=x^{2}-3 x+5$$

Graph the given function. Then find the slope or rate of change of the curve at the given value of \(x\), either manually, by zooming in, by using the TANGENT feature on your calculator, or numerically, as directed by your instructor. $$y=x^{2}-3 \quad \text { at } x=5$$

Evaluate each limit. $$\lim _{x \rightarrow-7} \frac{49-x^{2}}{x+7}$$

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. $$y=(x+3)(5 x-6)$$

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Derivative of a Constant Times a Power Function. $$y=x^{7}$$

Graph the given function. Then find the slope or rate of change of the curve at the given value of \(x\), either manually, by zooming in, by using the TANGENT feature on your calculator, or numerically, as directed by your instructor. $$y=3-\sqrt{x} \quad \text{ at } x=1$$

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

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. $$y=(4 x-1)(3 x+3)$$

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

Graph the given function. Then find the slope or rate of change of the curve at the given value of \(x\), either manually, by zooming in, by using the TANGENT feature on your calculator, or numerically, as directed by your instructor. $$y=\sqrt{x}-x \quad \text{ at } x=4$$

Find the derivative of each function. Check some by calculator. $$y=\frac{31.6}{1-2 x}$$

Evaluate each limit. $$\lim _{x \rightarrow 5} \frac{x^{2}-12 x+35}{5-x}$$

Find the derivative \(d x / d y\) of \(x\) with respect to \(y.\) $$x=y^{2}-7 y$$

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. $$y=x^{3}\left(8.24 x-6.24 x^{3}\right)$$

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

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

Evaluate each limit. $$\lim _{x \rightarrow 0} \frac{\sin x}{\tan x}$$

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

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Derivative of a Constant Times a Power Function. $$y=5.4 x^{3}$$