I was wondering if you could help me solve a couple of questions that I have to

Question

I was wondering if you could help me solve a couple of questions that I have to study for my calculus exam tomorrow. 1. Using implicit differentiation to find dy/dx (A) xe^y=x-y (B) x^2-4xy+y^2=4 2. Finding the tangent lines (A) find all points on the curve y=2x^3+3x^2-12x+3 where the tangent line is horizontal. (B) find an equation for the tangent line to the circle x^2+y^2=25 at point (4,3)
I was wondering if you could help me solve a couple of questions that I have to study for my calculus exam tomorrow. 1. Using implicit differentiation to find dy/dx (A) xe^y=x-y (B) x^2-4xy+y^2=4 2. Finding the tangent lines (A) find all points on the curve y=2x^3+3x^2-12x+3 where the tangent line is horizontal. (B) find an equation for the tangent line to the circle x^2+y^2=25 at point (4,3)
I was wondering if you could help me solve a couple of questions that I have to study for my calculus exam tomorrow. 1. Using implicit differentiation to find dy/dx (A) xe^y=x-y (B) x^2-4xy+y^2=4 2. Finding the tangent lines (A) find all points on the curve y=2x^3+3x^2-12x+3 where the tangent line is horizontal. (B) find an equation for the tangent line to the circle x^2+y^2=25 at point (4,3)

Explanation / Answer

1)

(A) xey=x-y

differentiate with respect to x on both sides. product rule:(uv)'=u'v +uv'

(1*ey)+(xeydy/dx) =1- (dy/dx)

(xeydy/dx) + (dy/dx)=1-ey

(xey +1) (dy/dx)=1-ey

(dy/dx)=(1-ey)/(xey +1)

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(B) x2-4xy+y2=4

differentiate with respect to x on both sides. product rule:(uv)'=u'v +uv'

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

2x -4y -4x(dy/dx)) +2y(dy/dx)=0

(-4x +2y)(dy/dx)=(4y -2x)

(dy/dx)=(4y -2x)/(-4x +2y)

(dy/dx)=(2y -x)/(-2x +y) or (dy/dx)=(-2y +x)/(2x -y)

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