Question

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.

Answer 1

`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`