math
posted by sam
Use implicit differentiation to find the equation of the tangent line to the curve xy3+xy=14 at the point (7,1) . The equation of this tangent line can be written in the form y=mx+b
i don't seem to no how to find m or b

JP
Take the derivative of both sides of the equation implicitly:
d/dx(xy^3+xy)=d/dx(14)
y^3+3xy^2*y'+y+xy'=0
At x=7, y=1. Thus, 1^3+3(7)(1^2)*y'+1+7y'=0
which yields y'=2/28=1/14
The slope of the tangent line is m which also is y'. Thus, you now have m=1/14. Since the point (7,1) is on the tangent line, then just plug these coordinates as well as the value of m into the equation y=mx+b and solve for b.
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