Chain Rule
The derivative of y=cos2(x3)
What is -6x2cos(x3)sin(x3)?
The derivative of f(x) = e2xcos(x)
What is f'(x) = -e2xsin(x)+2e2xcos(x) ?
The derivative with respect to x of x3+y3=4
What is y'=-x2/y2 ?
A plane is flying directly away from you at 500 mph at an altitude of 3 miles. How fast is the plan's distance from you increasing at the moment when the plane is flying over a point on the group 4 miles from you?
2xx'=2yy'
2(4)(500)=2(5)y'
y'=400mph
Suppose that F(x)=[g(f-1(x))]2,
f(2)=9, f'(2)=5, g(2)=3, g'(2)=-2
Find F'(9)
-60
The derivative of y=ln(cos5(3x4))
What is -60x3tan(3x4)?
The derivative of ln(2x3+5x2-3)
What is (6x2+10x) / (2x3+5x2-3) ?
dy/dx, if x=3+(x2+y2)1/2
What is ((x2+y2)1/2-x) / y ?
A cylindrical tank standing upright (with one circular base on the ground) has radius 1 meter. How fast does the water level in the tank drop when the water is being drained at 3 liters per second?
3/1000pi meters/second
The derivative of y=xx+1
What is (lnx + (x+1)/x) * xx+1 ?
The derivative of y=(1/5)sec-4(4+x3)
What is (-12/5)x2sec-4(4+x3)tan(4+x3)?
The derivative of
f(x) = ln((x2+1) / (x3-x))
What is
f'(x) = 2x/(x2+1) - (3x2-1)/(x3-x) ?
dy/dx, when cos2x+cos2y=cos(2x+2y)
A ladder 13 meters long rests on horizontal ground and leans against a vertical wall. The top of the ladder is being pulled up the wall at 0.1 m/s. How fast is the foot of the ladder approaching the wall when the foot of the ladder is 5m from the wall?
6/25 m/s
The derivative of y= ex / (x2+1)1/2
What is ex((x2+1)1/2 - 1/2(x2+1)1/2) / (x2+1) ?
The derivative of (sin(7x+ln(5x)))1/2
What is ((7x+1)cos(7x+ln(5x))) / (2x(sin(7x+ln(5x)))1/2)
The derivative of y=log5(-5x3-2)3
What is
(-45x2) / (ln5(-5x3-2)) ?
An equation of the line tangent to (x2+y2)3=8x2y2 at the point (-1,1)
A baseball diamond is a square 90 ft on a side. A player runs from first base to second base at 15 ft/sec. At what rate is the player's distance from third base decreasing when she is half way from first to second base?
3*sqrt(5) ft/s
The derivative of xsqrt(x)
What is dy/dx =((ln(x) + 2)*xsqrt(x)) / 2x1/2 ?
The derivative of (ln(x2+1)-tan-1(6x))10
What is
10( ln(x2+1) - tan-1(6x))9 (2x/(x2+1) - 6/(36x2+1)) ?
The derivative of y=3cos(3(x^4))
What is
-4x3 * 3cos(3(x^4)) * sin(3x4) * ln3 ?
All points (x,y) on the graph of x2/3+y2/3=8 that have tangent lines with slope=-1
What are (-8,-8) and (8,8) ?
A rotating beacon is located 2 miles out in the water. Let A be the point on the shore that is closest to the beacon. As the beacon rotates at 10 rev/min, the beam of light sweeps down the shore once each time it revolves. Assume that the shore is straight. How fast is the point where the beam hits the shore moving at an instant when the beam is lighting up a point 2 miles along the shore from the point A?
80pi mi/min
The derivative of (sin-1(x)) / (1+x)
What is (1 + x - (1-x2)1/2sin-1x) / ((1-x2)1/2(1+x)2) ?