An airplane flies at an altitude of 5 miles toward a point directly over an observer. Consider θ and x as shown in the following figure. Write θ as a function of x. ​ ​

Find the derivative of the function. ​ ​

Find the derivative of the function . ​

Suppose that an automobile’s velocity starting from rest is where v is measured in feet per second. Find the acceleration at 7 seconds. Round your answer to one decimal place. ​

Use implicit differentiation to find an equation of the tangent line to the ellipse  at . ​

Find the second derivative of the function . ​

Find the derivative of the function. ​ ​

Use logarithmic differentiation to find the derivative of . ​

Suppose the position function for a free-falling object on a certain planet is given by . A silver coin is dropped from the top of a building that is 1,366 feet tall. Find the instantaneous velocity of the coin when . ​

The radius of a right circular cylinder is and its height is , where t is time in seconds and the dimensions are in inches. Find the rate of change of the volume of the cylinder, V, with respect to time. ​