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What is the slope of the line tangent to the curve y+2 = (x^2/2) - 2siny at the point (2,0)?

User Latrisha
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1 Answer

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implicit differentiation
take the deritivive with respect to x

first
y+2=(1/2)(x^2)-2sin(y)
take deritivitive
dy/dx+0=1/2(2x)-2(cosy)dy/dx
dy/dx=x-2cosy(dy/dx)
add 2cosy(dy/dx) to both sides
dy/dx+2cosy(dy/dx) =x
undistribute dy/dx
dy/dx(1+2cos(y))=x
divide both sides by 1+2cos(y)

(dy)/(dx)= (x)/(1+2cos(y))
at (2,0)
x=2
y=0


(dy)/(dx)= (2)/(1+2cos(0))

(dy)/(dx)= (2)/(1+2(1))

(dy)/(dx)= (2)/(1+2)

(dy)/(dx)= (2)/(3)
the slope is 2/3
User Seleena
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