Find the derivative of (3x + 4)^{2x} Method 1: use implicit differentiation and logarithmic differentiation: Write as an equation and “simplify” until the equation is in a format in which you know how to take the derivative: y = (3x + 4)^{2x} ln(y) = ln(3x + 4)^{2x} ln(y) = 2x · ln(3x + 4) Now that we have an equation where we know how to take the derivative of both sides, we can take the derivative using implicit differentiation: \frac{1}{y} y′ = 2ln(3x + 4) + 2x(\frac{1}{3x+4})3 y′ = y[2ln(3x+4)+ \frac{6x}{3x+4}] = (3x+4)^{2x}[2ln(3x+4)+ \frac{6x}{3x+4}] Method 2: Use logarithmic differentiation directly: (3x + 4)^{2x} = e^{ln(3x+4)^{2x}} Thus [(3x + 4)^{2x}]′ = [e^{ln(3x+4)^{2x}}]′ = [e^{2x·ln(3x+4)}]′ 1.) Suppose car A is 110 miles north of an intersection and traveling south at 50 mph. Suppose car B is 100 miles east of the same intersection and traveling west at 20 mph. 1a.) At what rate are the cars approaching each other after 1 hour? 1b.) After 3 hours?

2.) Suppose the distance between two planes must be maintained at 10 miles. Suppose plane W is north of a radio tower and moving south while plane G is east of the same radio tower. If plane G is moving east at 1 mile/second, how fast should plane W be moving when plane G is 6 miles from the radio tower?

3.) A water tank has the shape of an inverted circular cone with base radius 12 m and height 4 m. Suppose water is leaking out of the cone at a rate of 5 m^{3}/min. while water is being pumped into the cone at a rate of 9 m^{3}/min. Find the rate at which the water level is rising when the water is 1.5 m deep.

HW7 3.7) 31, 39 3.8) 11, 15, 17, 19, 21, 25, 29, 31, 41, 43, 47, 49 3.10) 2-38 even (ie, all even problems)

3/8: Quiz 5: 3.6, 3.7, 3.8