find the indicated derivatives, using the following information:\n$f(3)=-1$\n$g(3)=6$\n$f(3)=1$\n$g(3)=5$\n$(…

Explanation:

Step1: Use the product rule

The product rule states that ((fg)'(x)=f'(x)g(x)+f(x)g'(x)). Substitute (x = 3): ((fg)'(3)=f'(3)g(3)+f(3)g'(3)) Given (f(3)=-1), (f'(3)=1), (g(3)=6), (g'(3)=5) ((fg)'(3)=(1)\times(6)+(-1)\times(5))

Step2: Calculate the value

((fg)'(3)=6 - 5)

Step3: Use the quotient rule

The quotient rule states that (\left(\frac{f}{g}\right)'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^{2}}) Substitute (x = 3): (\left(\frac{f}{g}\right)'(3)=\frac{f'(3)g(3)-f(3)g'(3)}{[g(3)]^{2}}) (\left(\frac{f}{g}\right)'(3)=\frac{(1)\times(6)-(-1)\times(5)}{6^{2}})

Step4: Calculate the value

(\left(\frac{f}{g}\right)'(3)=\frac{6 + 5}{36}=\frac{11}{36})

Answer:

((fg)'(3)=1) (\left(\frac{f}{g}\right)'(3)=\frac{11}{36})