If 5x^2+5x+xy=2 and y(2)= -14 find y'(2) by implicit differentiation

Answer 1

5 x² + 5 x + x y = 2

Differentiate both sides of the equation with respect to x.

Apply the sum / difference rule:​ ( f ± g )′ = f ′ ± g′

( 5 x² )′ + ( 5 x )′ + ( x y )′ = 2′

5 ∙ 2 x + 5 + ( x y )′ = 0

10 x + 5 + ( x y )′ = 0
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For ( x y )′ apply the product rule:

( f ∙ g )′ = f ′ ∙ g + f ∙ g′

( x y )′ = x ′ ∙ y + x ∙ y′ (x)

( x y )′ = 1 ∙ y + x ∙ y′ (x)

( x y )′ = y + x ∙ y′ (x)
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10 x + 5 + y + x ∙ y′ (x) = 0

y(2) = -14 mean x = 2 , y = - 14

10 ∙ 2 + 5 + ( - 14 ) + 2 ∙ y′ ( 2 ) = 0

20 + 5 - 14 + 2 ∙ y′ ( 2 ) = 0

11 + 2 ∙ y′ ( 2 ) = 0

2 ∙ y′ ( 2 ) = - 11

y′ ( 2 ) = - 11 / 2