How do you use implicit differentiation to find the slope of the curve given #12(x^2+y^2)=25xy# at (3,4)?

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Answer 1 · 2017-07-30T14:46:29

Please see the explanation.

Given: #12(x^2+y^2)=25xy#

Use the distributive property:

#12x^2+12y^2=25xy#

Differentiate each term:

#d/dx12x^2+d/dx12y^2=d/dx25xy#

The first term is just the power rule:

#24x+d/dx12y^2=d/dx25xy#

The second term requires the chain rule:

#24x+24ydy/dx=d/dx25xy#

The last term requires the use of the product rule:

#24x+24ydy/dx=25y+ 25xdy/dx#

Collect all of the terms containing #dy/dx# on the left and the other terms on the right:

#24ydy/dx-25xdy/dx=25y-24x #

Factor out #dy/dx#:

#(24y-25x)dy/dx=25y-24x #

Divide both sides by the leading factor on the left:

#dy/dx=(25y-24x)/(24y-25x)#

The slope, m, of the tangent line is the above evaluated at #(3,4)#:

#m = (25(4)-24(3))/(24(4)-25(3))#

#m = 4/3#