Chapter 23

Find the derivative. $$\frac{d}{d x}\left(x^{2}-1\right)$$

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 Sum. $$y=a x^{5}-5 b x^{3}$$

Write the differential \(d y\) for each function. $$y=x^{2}+2 x$$

$$y=x \sqrt{x+1} \sqrt[3]{x}$$

Find the slope of the tangent to the curve \(y=1 /(x+1)\) at \(x=2\).

Find the derivative. $$D_{x}(7-5 x)$$

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 Sum. $$y=\frac{x^{2}}{2}-\frac{x^{7}}{7}$$

Write the differential \(d y\) for each function. $$y=\frac{x-1}{x+1}$$

Find the derivative of each function.. $$y=\frac{x}{x+2}$$

When the Limit Is an Expression $$\lim _{d \rightarrow 0} \frac{(x+d)^{2}-x^{2}}{x^{2}(x+d)}$$

If \(y=\left(x^{2}-x\right)^{3},\) find \(y^{\prime}(3)\).

Find the derivative. $$D_{x}\left(x^{2}\right)$$

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 Sum. $$y=\frac{x^{3}}{1.75}+\frac{x^{2}}{2.84}$$

Write the differential \(d y\) for each function. $$y=\left(2-3 x^{2}\right)^{3}$$

Find the derivative of each function.. $$y=\frac{x}{x^{2}+1}$$

When the Limit Is an Expression $$\lim _{d \rightarrow 0} \frac{(x+d)^{2}-x^{2}}{d}$$

If \(f(x)=\sqrt[3]{2 x}+(2 x)^{2 / 3},\) find \(f^{\prime}(4)\).

Find the derivative. $$D(3 x+2)$$

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 Sum. $$y=2 x^{3 / 4}+4 x^{-1 / 4}$$

Write the differential \(d y\) for each function. $$y=x^{3}+3 x$$

Find the derivative of each function.. $$y=\frac{x^{2}}{4-x^{2}}$$

When the Limit Is an Expression $$\lim _{d \rightarrow 0} \frac{3(x+d)-3 x}{d}$$

We will see in a later chapter that the acceleration of a point is the rate of change of the velocity of the point. If the velocity of the arm of an industrial robot is given by \(v=3.45\left(t^{2}+2\right)^{2} \mathrm{ft} / \mathrm{s},\) where \(t\) is the time in seconds, take the derivative of this velocity to find the acceleration, and evaluate it at \(t=1.00 \mathrm{s}.\)

Find the derivative. $$D\left(x^{2}-1\right)$$

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 Sum. $$y=\frac{2}{x}-\frac{3}{x^{2}}$$

Write the differential \(d y\) for each function. $$y=\sqrt{1-2 x}$$

Find the derivative of each function.. $$y=\frac{x-1}{x+1}$$

When the Limit Is an Expression $$\lim _{d \rightarrow 0} \frac{[2(x+d)+5]-(2 x+5)}{d}$$

Write the differential \(d y\) in terms of \(x, y,\) and \(d x\) for each implicit relation. $$3 x^{2}-2 x y+2 y^{2}=3$$

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 Sum. $$y=2 x^{4 / 3}-3 x^{2 / 3}$$

Find the derivative of each function.. $$y=\frac{x+2}{x-3}$$

Writing: We find the derivative by the delta method by first finding \(\Delta y\) divided by \(\Delta x\) and then letting \(\Delta x\) approach zero. Explain in your own words why this doesn't give division by zero, causing us to junk the whole calculation.

Write the differential \(d y\) in terms of \(x, y,\) and \(d x\) for each implicit relation. $$x^{3}+2 y^{3}=5$$

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

Find the derivative of each function.. $$y=\frac{2 x-1}{(x-1)^{2}}$$

When the Limit Is an Expression $$\lim _{d \rightarrow 0} \frac{\left[(x+d)^{2}+1\right]-\left(x^{2}+1\right)}{d}$$

Write the differential \(d y\) in terms of \(x, y,\) and \(d x\) for each implicit relation. $$2 x^{2}+3 x y+4 y^{2}=20$$

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Other Symbols for the Derivative. $$\text { If } y=2 x^{3}-3, \text { find } y^{\prime}$$

Find the derivative of each function.. $$y=\frac{x^{1 / 2}}{x^{1 / 2}+1}$$

When the Limit Is an Expression $$\lim _{d \rightarrow 0} \frac{(x+d)^{3}-x^{3}}{d}$$

Write the differential \(d y\) in terms of \(x, y,\) and \(d x\) for each implicit relation. $$2 \sqrt{x}+3 \sqrt{y}=4$$

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Other Symbols for the Derivative. $$\text { If } f(x)=7-4 x^{2}, \text { find } f^{\prime}(x)$$

Find the derivative of each function.. $$s=\sqrt{\frac{t-1}{t+1}}$$

Evaluate each expression. $$\frac{d}{d x}\left(3 x^{5}+2 x\right)$$

Find the derivative of each function.. $$w=\frac{z}{\sqrt{z^{2}-a^{2}}}$$

When the Limit Is an Expression $$\lim _{d \rightarrow 0} \frac{(x+d)^{2}-2(x+d)-x^{2}+2 x}{d}$$

Evaluate each expression. $$\frac{d}{d x}\left(2.5 x^{2}-1\right)$$

Find the derivative of each function.. $$v=\sqrt{\frac{1+2 t}{1-2 t}}$$

When the Limit Is an Expression $$\lim _{d \rightarrow 0} \frac{\frac{7}{x+d}-\frac{7}{x}}{d}$$

Find the slope of the tangent to the curve \(y=\sqrt{16+3 x} / x\) at \(x=3\)