13. -/1 points if (f(x)=\frac{x^{2}}{2 + x}), find (f(2)). (f(2)=) blank resources read it watch it submit…

Explanation:

Step1: Find the first - derivative of (f(x)=\frac{x^{2}}{2 + x}) using the quotient rule

The quotient rule states that if (y=\frac{u}{v}), then (y'=\frac{u'v - uv'}{v^{2}}). Here, (u = x^{2}), (u'=2x), (v=2 + x), (v' = 1). So (f'(x)=\frac{2x(2 + x)-x^{2}(1)}{(2 + x)^{2}}=\frac{4x+2x^{2}-x^{2}}{(2 + x)^{2}}=\frac{x^{2}+4x}{(2 + x)^{2}}).

Step2: Find the second - derivative of (f(x)) using the quotient rule again

Let (u=x^{2}+4x), (u'=2x + 4), (v=(2 + x)^{2}), (v'=2(2 + x)). Then (f''(x)=\frac{(2x + 4)(2 + x)^{2}-(x^{2}+4x)\times2(2 + x)}{(2 + x)^{4}}=\frac{(2x + 4)(2 + x)-2(x^{2}+4x)}{(2 + x)^{3}}=\frac{4x+2x^{2}+8 + 4x-2x^{2}-8x}{(2 + x)^{3}}=\frac{8}{(2 + x)^{3}}).

Step3: Evaluate (f''(2))

Substitute (x = 2) into (f''(x)): (f''(2)=\frac{8}{(2 + 2)^{3}}=\frac{8}{64}=\frac{1}{8}).

Answer:

(\frac{1}{8})

Explanation for 14(a):

Step1: Use the product rule ((fg)'(x)=f'(x)g(x)+f(x)g'(x))

We know that (f(5) = 1), (f'(5)=5), (g(5)=-2), and (g'(5)=3). Substitute these values into the product - rule formula: ((fg)'(5)=f'(5)g(5)+f(5)g'(5)).

Step2: Calculate the value

((fg)'(5)=5\times(-2)+1\times3=-10 + 3=-7).

Answer for 14(a):

(-7)

Explanation for 14(b):

Step1: Use the quotient rule ((\frac{f}{g})'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{g^{2}(x)})

Substitute (x = 5), (f(5) = 1), (f'(5)=5), (g(5)=-2), and (g'(5)=3) into the quotient - rule formula: ((\frac{f}{g})'(5)=\frac{f'(5)g(5)-f(5)g'(5)}{g^{2}(5)}).

Step2: Calculate the value

((\frac{f}{g})'(5)=\frac{5\times(-2)-1\times3}{(-2)^{2}}=\frac{-10 - 3}{4}=-\frac{13}{4}).

Answer for 14(b):

(-\frac{13}{4})

Explanation for 14(c):

Step1: Use the quotient rule ((\frac{g}{f})'(x)=\frac{g'(x)f(x)-g(x)f'(x)}{f^{2}(x)})

Substitute (x = 5), (f(5) = 1), (f'(5)=5), (g(5)=-2), and (g'(5)=3) into the quotient - rule formula: ((\frac{g}{f})'(5)=\frac{g'(5)f(5)-g(5)f'(5)}{f^{2}(5)}).

Step2: Calculate the value

((\frac{g}{f})'(5)=\frac{3\times1-(-2)\times5}{1^{2}}=\frac{3 + 10}{1}=13).

Answer for 14(c):

(13)