Find dy/dx by implicit differentiation. sin x + cos y = sin x cos y x^2y+xy^2 =

Question

Find dy/dx by implicit differentiation. sin x + cos y = sin x cos y x^2y+xy^2 = 3x Find the equation of the normal line to the curve x^2 + 4xy+y^2 = 13at the point (2.1). Gravel is being dumped from a convey or belt at a rate of 30 cubic feet per minute, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 feet high? An aircraft is flying horizontally at a constant height of 5000 ft above a fixed observation point P. At a certain instant the angle of elevation theta is 30degree and decreasing, and the speed of the aircraft is 300 mi/hr. Determine how fast theta is decreasing at this instant (in rad/sec) and how fast the distance between the aircraft and the observation point is changing at this instant.

Explanation / Answer

4a)

x^2y +xy^2 = 3x

differentiate

x^2 * dy/dx + 2xy + x*2ydy/dx +y^2 = 3

dy/dx(x^2+2xy) = 3-2xy-y^2

dy/dx = (3-2xy-y^2)/(x^2+2xy) ---------ANSWER

4b)

sinx +cosy = sinx cosy

differentiate

cosx -siny * dy/dx = sinx * (-siny * dy/dx) + cosy *cosx

dy/dx *siny (sinx -cosx) = cosy*cosx

dy/dx = cot y * cosx /(sinx-cosx ) --------ANSWER

5)

x^2+4xy +y^2 = 13

differentuate

2x+ 4x*dy/dx +4y +2y*dy/dx = 0

put x= 2 , y= 1

2*2 +(4*2+2*1) dy/dx +4*1 = 0

dy/dx = -8/10 =m1

we have m1*m2 = -1

-8/10 ( m2) = -1 =>m2 = 10/8

eq of normal : y= 10x/8 +c

1 = 10*2/8 +c

c = -3/2

so eq of normal is y = 10x/8-3/2 ------ANSWER

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