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) ?

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

1 Answer
Feb 22, 2018

y= -3x+22

Explanation:

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