Let $y=\sqrt{x}$. Sketch a diagram showing the line segments with lengths $d x, d y$, and $\Delta y$.

Let $y=\sqrt{x}$. Find the differential $d y$.

Find $\frac{d y}{d x}$ by implicit differentiation.

$$2 x e^x+2 y e^y=4$$

Find the derivative of the function.

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

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.

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

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

Differentiate the function.

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

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.

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

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

Differentiate the function.

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

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.