My Math Forum Equation of the tangent line

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 February 6th, 2012, 06:06 AM #1 Member   Joined: Jan 2012 Posts: 82 Thanks: 0 Equation of the tangent line Hi! I need to find dy/dx explicitly (i.e. in the form dy/dx=g(x,y) if x^3+y^2+2y=2. Then I need to determine the equation of the tangent to the curve represented by the above equation at the point (-1,1). Can anyone help? Thanks
 February 6th, 2012, 06:30 AM #2 Global Moderator     Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4 Re: Equation of the tangent line This is called "implicit differentiation", since you haven't/won't solve for y explicitly before differentiating. x^3+y^2+2y=2 We are going to use the chain rule on EVERY TERM. I will write dx/dx, just so you can see that the chain rule is always in play. However, dx/dx is just equal to 1, so we will make the appropriate simplification... (x^3) ' + (y^2) ' + (2y) ' = (2) ' ......here I have just put a ' after every term to indicate that we are differentiating. (3x^2 dx/dx) + (2y dy/dx) + (2 dy/dx) = 0 ... chain rule, power rule, and/or constant rule Factor out the dy/dx and simplify... (3x^2) + (dy/dx)*(2y + 2) = 0 Solve for dy/dx... (dy/dx)*(2y + 2) = -3x^2 $\dfrac{dy}{dx}= \dfrac{-3x^2}{2y + 2}$ after dividing. Plug in x = -1, y = 1 to THIS formula to get the slope at the point (-1, 1). Then use point slope formula to get an equation of the tangent line.
 February 6th, 2012, 06:36 AM #3 Member   Joined: Jan 2012 Posts: 82 Thanks: 0 Re: Equation of the tangent line Thanks, I thought that. Can you please help me with the point slope formula.
 February 6th, 2012, 07:05 AM #4 Member   Joined: Jan 2012 Posts: 82 Thanks: 0 Re: Equation of the tangent line Actually Is the following correct, plug in x=-1 and y=1, i get the slope m = -3/4 using the point slope fomula i get y-y1=m(x-x1) y-1=-3/4(x--1) y=-3/4x+1/4
 February 6th, 2012, 07:24 AM #5 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Equation of the tangent line Yes, that is correct.
 February 9th, 2012, 01:29 AM #6 Member   Joined: Jan 2012 Posts: 82 Thanks: 0 Re: Equation of the tangent line Thank you for your help

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