differentiation problems

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michy123

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Dec 15, 2005
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I was wondering if anyone can help me solve these differentiation problems, i dont even know where to start on them, if someone could be kind enought to explain step by step it would be really apprecitated.

1. if 2x^3+ 3xy + e^y= 6, find the exact value of dy/dx when x=0

2. find the equations of the tangent and normal at (1, -1) given the curve 4x^2 + 2xy- xy^3=0

3. show that the cirves y= -sinx/2 are orthogonal

Thank you in advance any help is appreciated:)
 
"if 2x^3+ 3xy + e^y= 6, find the exact value of dy/dx when x=0"


First, evaluate the the y value at x=0

e^y=6

y=ln(6)

Implicit differentiation: you're integrating y with respect to x.


6x^2+3(xdy/dx+y)+e^ydy/dx=0

(3x+e^y)dy/dx=-6x^2-3y


dy/dx=(-6x^2-3y)/(3x+e^y)


Now, substitute (0,ln(6)) for (x,y) and you have your answer.


For the second problem, the slope of the tangent line is dy/dx. See if you can figure it out.


For the third problem, orthogonal means perpendicular, but I don't understand the question. Shouldn't you have two curves?
 
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