homework3: problem 26\n(1 point)\ngiven that:\n$f(-5)=4,f(-5)=4,$\n$g(-5)=1,g(-5)=3,$\ncalculate the…

homework3: problem 26\n(1 point)\ngiven that:\n$f(-5)=4,f(-5)=4,$\n$g(-5)=1,g(-5)=3,$\ncalculate the following:\n$(fg)(-5)=\\square$\n$(f/g)(-5)=\\square$\n$(g/f)(-5)=\\square$\nnote: you can earn partial credit on this problem.\npreview my answers submit answers\nyou have attempted this problem 0 times.\nyou have unlimited attempts remaining.\nemail instructor

homework3: problem 26\n(1 point)\ngiven that:\n$f(-5)=4,f(-5)=4,$\n$g(-5)=1,g(-5)=3,$\ncalculate the following:\n$(fg)(-5)=\\square$\n$(f/g)(-5)=\\square$\n$(g/f)(-5)=\\square$\nnote: you can earn partial credit on this problem.\npreview my answers submit answers\nyou have attempted this problem 0 times.\nyou have unlimited attempts remaining.\nemail instructor

Answer

Explanation:

Step1: Use the product rule

The product rule states that ((fg)^\prime(x)=f^\prime(x)g(x)+f(x)g^\prime(x)). Substitute (x = - 5): ((fg)^\prime(-5)=f^\prime(-5)g(-5)+f(-5)g^\prime(-5)) (=4\times1 + 4\times3) (=4 + 12) (=16)

Step2: Use the quotient rule

The quotient rule states that ((\frac{f}{g})^\prime(x)=\frac{f^\prime(x)g(x)-f(x)g^\prime(x)}{g(x)^2}). Substitute (x=-5): ((\frac{f}{g})^\prime(-5)=\frac{f^\prime(-5)g(-5)-f(-5)g^\prime(-5)}{g(-5)^2}) (=\frac{4\times1-4\times3}{1^2}) (=\frac{4 - 12}{1}) (=-8)

Step3: Use the quotient rule again

The quotient rule states that ((\frac{g}{f})^\prime(x)=\frac{g^\prime(x)f(x)-g(x)f^\prime(x)}{f(x)^2}). Substitute (x = - 5): ((\frac{g}{f})^\prime(-5)=\frac{g^\prime(-5)f(-5)-g(-5)f^\prime(-5)}{f(-5)^2}) (=\frac{3\times4-1\times4}{4^2}) (=\frac{12 - 4}{16}) (=\frac{8}{16}) (=\frac{1}{2})

Answer:

((fg)^\prime(-5)=16) ((f/g)^\prime(-5)=-8) ((g/f)^\prime(-5)=\frac{1}{2})