Let $y=\sqrt{x}$. Evaluate $d y$ and $\Delta y$ if $x=1$ and $d x=\Delta x=1$.

Find the derivative of the function.

$$g(u)=\left(\dfrac{u^3-1}{u^3+1}\right)^8$$

Differentiate the function.

$$f(x)=x^{2.4}+e^{2.4}$$

Find an equation of the tangent line to the curve $y=\ln \left(x^3-7\right)$ at the point (2,0).

Differentiate the function.

$$F(t)=\dfrac{A t}{B t^2+C t^3}$$

Consider $f(x)=x^{5 / 2} e^x$. Find $f^{\prime}(x)$ and $f^{\prime \prime}(x)$.

Consider $y=\ln \sin x-\frac{1}{2} \sin ^2 x$. Calculate $y^{\prime}$.

Find the derivative of $y=\cos ^{-1}\left(\sin ^{-1} t\right)$. Simplify where possible.

Find an equation of the tangent line to the graph of the given function at the specified point.

$$f(x)=\cos x-\sin x, \quad(\pi,-1)$$

Differentiate the function.

$$f(x)=3^x x^3$$

Find the derivative of $y=\dfrac{e^u-e^{-u}}{e^u+e^{-u}}$.

The table gives the population of the world in the 20th century.

$$\begin{array}{|c|c||c|c|} \hline \text { Year } & \begin{array}{c} \text { Population } \\ \text { (in millions) } \end{array} & \text { Year } & \begin{array}{c} \text { Population } \\ \text { (in millions) } \end{array} \\ \hline 1900 & 1650 & 1960 & 3040 \\ 1910 & 1750 & 1970 & 3710 \\ 1920 & 1860 & 1980 & 4450 \\ 1930 & 2070 & 1990 & 5280 \\ 1940 & 2300 & 2000 & 6080 \\ 1950 & 2560 & & \\ \hline \end{array}$$

Estimate the rate of growth in 1985.

Verify $1 /(1+2 x)^4 \approx 1-8 x$ at $a=0$. Then determine the values of $x$ for which the linear approximation is accurate to within 0.1.

The table gives the population of the world in the 20th century.

$$\begin{array}{|c|c||c|c|} \hline \text { Year } & \begin{array}{c} \text { Population } \\ \text { (in millions) } \end{array} & \text { Year } & \begin{array}{c} \text { Population } \\ \text { (in millions) } \end{array} \\ \hline 1900 & 1650 & 1960 & 3040 \\ 1910 & 1750 & 1970 & 3710 \\ 1920 & 1860 & 1980 & 4450 \\ 1930 & 2070 & 1990 & 5280 \\ 1940 & 2300 & 2000 & 6080 \\ 1950 & 2560 & & \\ \hline \end{array}$$

Use a model for the rate of population growth in the 20th century to estimate the rates of growth in 1920 and 1980. Compare with the estimates by averaging the slopes of two secant lines.