Use implicit differentiation to find an equation of the tangent line to the curv

Question

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x^2 + 2xy - y^2 + x = 11, (4, 9) (hyperbola) y = Find y" by implicit differentiation. 5x^2 + y^2 = 8 y" = Find dy/dx by implicit differentiation. 1 + 4x - sin(xy^2) dy/dx = The cost function for production of a commodity is C(x) = 369 + 24x - 0.06x^2 + 0.0007x^2. (a) Find C'(100). Interpret C'(100). This is the cost of making 100 items. This is the amount of time, in minutes, it takes to produce 100 items. This is the number of items that must be produced before the costs reach 100. This is the rate at which costs are increasing with respect to the production level when x = 100. This is the rate at which the production level is decreasing with respect to the cost when x = 100. (b) Find the actual cost of producing the 101st item. (Round your answer to the nearest cent.) exist

Explanation / Answer

2.

5x^2 + y^2 = 8

Using differentiation:

2*5*x + 2*y*y' = 0

10x + 2*y*y' = 0

Again using differentiation:

10*1 + 2*y*y'' + 2*y'*y' = 0

2*y*y'' = -(10 + 2*(y')^2)

y'' = -2*(5 + (y')^2)/(2y)

y'' = -*(5 + (y')^2)/y

y'' = -5/y + (y')^2/y

3.

1 + 4x = sin (xy^2)

Using differentiation

0 + 4*1 = cos (x*y^2)*[1*y^2 + 2*y*y']

2*y*y'*cos (xy^2) = [4 - y^2*cos (xy^2)]

y' = [4 - y^2*cos (xy^2)]/[2y*cos (xy^2)]

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