Find g'(2).?

Question

(g(x))^2+14x=x^2g(x)+49, and g(2)=7. Thanks!.

Answer 1

[g(x)]² +14x = x² g(x) + 49 

g² +14x = x² g + 49

Derivative the function implicitly with respect to x .

 2g g' +14 = 2xg + x² g' 

2g g' -  x² g' = 2xg -14

g' ( 2g - x² ) =  2xg -14

g' = (2xg -14) / ( 2g - x² )

g'(2) = (2(2)g(2) -14) / ( 2g(2) - 2² )

g'(2) = (4g(2) -14) / ( 2g(2) - 4 )

Given g(2) = 7

g'(2) = (4*7 -14) / ( 2*7 - 4 )

g'(2) = (28 -14) / ( 14 - 4 )

g'(2) = 14 / 10

g'(2) = 7/5 .

So the g'(2) = 7/5 .