Consider $y=\dfrac{\left(x^2+1\right)^4}{(2 x+1)^3(3 x-1)^5}$. Calculate $y^{\prime}$.

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

Use logarithmic differentiation to find the derivative of the function.

$$y=\sqrt{\dfrac{x-1}{x^4+1}}$$

Find the derivative of $y=x \sin ^{-1} x+\sqrt{1-x^2}$. Simplify where possible.

Find $f^{\prime}(x)$ and $f^{\prime \prime}(x)$.

$$f(x)=\dfrac{x^2}{1+e^x}$$

Differentiate the function.

$$D(t)=\dfrac{1+16 t^2}{(4 t)^3}$$

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

$$x^2+y^2=\left(2 x^2+2 y^2-x\right)^2, \quad\left(0, \frac{1}{2}\right) \quad (\text{cardioid})$$

Find the derivative of the function.

$$J(\theta)=\tan ^2(n \theta)$$

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

Consider $x e^y=y-1$. Calculate $y^{\prime}$.

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.

The table shows the population of Canada (in millions) at the end of the given year.

Use a linear approximation to estimate the population on December 31, 2009. Use another linear approximation to predict the population in 2025.

Differentiate the function.

$$f(x)=\dfrac{\ln x}{x^2}$$

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.