help on implict differentiation?

Question

(g(x))^2+11x=x^2g(x) +22
and that g(2)=4, find g'(2).
g'(2)____________

Answer 1

Take the derivative with respect to x on both sides of the equation.

The derivative of g(x)^2 is 2*g(x)*g'(x) using the chain rule. The derivative of x^2*g(x) is x^2*g'(x) + 2x * g(x) using the multiplication rule. So we get

2*g(x)*g'(x) + 11 = x^2 * g'(x) + 2x * g(x)

Now solve for g'(x):

g'(x)(2*g(x) - x^2) = 2x * g(x) - 11
g'(x) = (2x * g(x) - 11) / (2*g(x) - x^2)

Plugging in x=2 and substituting 4 for g(2) gives us

g'(2) = (2*2*4-11) / (2*4 - 2^2)
g'(2) = (16 - 11) / (8 - 4)
g'(2) = 5/4.

Answer 2

Differentiate implicitly at x = 2,
2g(2)g'(2) + 11 = 2(2)g(2) + 2^2g'(2)
Simplify,
8g'(2) + 11 = 16 + 4g'(2)
Solve for g'(2),
g'(2) = 5/4

Answer 3

(g(x))^2 + 11x = x^2g(x) + 22

differentiate

2g(x) g'(x) + 11 = x^2g'(x) + 2x g(x)

put x = 2

2g(2) g'(2) + 11 = 4 g'(2) + 4g(2)

but g(2) =4

2(4) g'(2) + 11 = 4 g'(2) + 4(4)

8 g'(2) + 11 = 4g'(2) + 16

4g'(2) = 5

g'(2) = 5/4

Answer 4

2g(x)*g´(x) +11=2x*g(x) +x^2*g´(x)
so g´(x) = [2x*g(x)-11]/[2 g(x)-x^2]
g´(2) = (16-11)/(8-4)=5/4

Answer 5

2.g(x).g'(x)+11 = 2.x.g(x)+x^2.g'(x)+0
~ g'(x).[2g(x)-x^2] = 2.x.g(x)-11
~ g'(2)[2.4-4] = 2.2.4-11
~4.g'(2) = 16-11= 5
~ g'(2) = 5/4