assume that ( f(x) ) is differentiable. find an expression for the derivative of ( y ) at ( x = 4 )…

assume that ( f(x) ) is differentiable. find an expression for the derivative of ( y ) at ( x = 4 ), assuming that ( f(4)=-3 ) and ( f^{prime}(4)=3 )\n( y=\frac{f(x)+1}{f(x)+x} )\nthe derivative of ( y ) at ( x = 4 ) is ( square ).

assume that ( f(x) ) is differentiable. find an expression for the derivative of ( y ) at ( x = 4 ), assuming that ( f(4)=-3 ) and ( f^{prime}(4)=3 )\n( y=\frac{f(x)+1}{f(x)+x} )\nthe derivative of ( y ) at ( x = 4 ) is ( square ).

Answer

Explanation:

Step1: Apply quotient rule

The quotient rule states that if (y = \frac{u}{v}), then (y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}). Here, (u = f(x)+1), so (u^\prime=f^\prime(x)); (v = f(x)+x), so (v^\prime=f^\prime(x)+1). Then (y^\prime=\frac{f^\prime(x)(f(x)+x)-(f(x)+1)(f^\prime(x)+1)}{(f(x)+x)^{2}}).

Step2: Substitute (x = 4)

Given (f(4)=- 3) and (f^\prime(4)=3). First, find (u(4)=f(4)+1=-3 + 1=-2), (v(4)=f(4)+4=-3 + 4 = 1), (u^\prime(4)=f^\prime(4)=3), (v^\prime(4)=f^\prime(4)+1=3 + 1=4). Then (y^\prime(4)=\frac{3\times1-(-2)\times4}{1^{2}}). Calculate the numerator: (3\times1-(-2)\times4=3 + 8=11).

Answer:

(11)