If `4x^2+2x+xy=4 ` and `y(4)=-17` , find `y'(4)` using implicit differentiation.Implicit differenation. If 4x^2+2x+xy=4 and y(4)=-17, find y'(4) by impliciy differeation

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lemjay eNotes educator| Certified Educator


To start, take the derivative of both sides.

`d/(dx) (4x^2+2x+xy)=d/(dx)4`

`d/(dx) 4x^2 + d/(dx)2x + d/(dx)xy = d/(dx)4`

Let's take the derivative of each terms separately.

For the first and second term at the left side, apply the power formula of derivatives which is `d/(dx) x^n=nx^(n-1)` .   

>> `d/(dx)4x^2 = 4*2x=8x`

>> `d/(dx)2x=2*x^0=2*1=2`

For the third term, apply the product rule: `d/(dx)(u*v)=uv'+vu'` .

>> `d/(dx) xy = xy' + y*1 = xy'+y`

For the right side, note that the derivative of a constant is zero `(d/(dx)c=0 )` .

>> `d/(dx)4 =0`

So when taking the derivative of both sides of the equation yields,

`d/(dx) 4x^2 + d/(dx)2x + d/(dx)xy = d/(dx)4`

          `8x + 2 + xy'+y=0`

Next, isolate y'.

`xy' = -8x-y-2`



Base on the given condition y(4)=-17, substitute x=4 and y=-17 to y'.

`y'=-(8*4+(-17)+2)/4=-(32-17+2)/4 = -17/4`

Hence, `y'(4)=-17/4` .