Use implicit differentiation to find an equation of the tangent line to the curve at the given point. #x^2 + 2xy − y^2 + x = 51# (5, 7) (hyperbola) ?

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I would need a step by step explanation to complete this math problem and thanks for your help in advance.

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Answer 1 · 2018-02-22T10:48:46

#y= -3x+22#

Differentiate all the terms individually

#f(x): x^2+2xy-y^2+x=51#
#f'(x): 2x+2x(dy)/(dx)+2y-2y(dy)/(dx)+1=0#

Factorise out / isolate the #dy/dx# on one side
#2x(dy)/(dx)-2y(dy)/(dx)=-2x-2y-1#

#(dy)/(dx)(2x-2y)=-2x-2y-1#

#(dy)/(dx)=(-2x-2y-1)/(2x-2y)#

Find the gradient at (5,7) by subbing these values into your differential equation

#(dy)/(dx)=25/4#

Equation of tangent
#y-y_1 = m(x-x_1))#
#y-7 = 25/4(x-5)#

#y-7 = 25/4x-125/4#
#y= 25/4x-97/4#