let (f(5)=5), (f(5)= - 2), (g(5)= - 7), and (g(5)=2). calculate the following.\na. ((f + g)(5)=)\nb. ((f…

Explanation:

Step1: Use sum - rule of derivatives

The sum - rule states that $(f + g)'(x)=f'(x)+g'(x)$. So, $(f + g)'(5)=f'(5)+g'(5)$. Substitute $f'(5)= - 2$ and $g'(5)=2$ into the formula: $(f + g)'(5)=-2 + 2$

Step2: Calculate the result

$-2+2 = 0$

Step3: Use difference - rule of derivatives

The difference - rule states that $(f - g)'(x)=f'(x)-g'(x)$. So, $(f - g)'(5)=f'(5)-g'(5)$. Substitute $f'(5)= - 2$ and $g'(5)=2$ into the formula: $(f - g)'(5)=-2-2=-4$

Step4: Use product - rule of derivatives

The product - rule states that $(f\cdot g)'(x)=f'(x)g(x)+f(x)g'(x)$. So, $(f\cdot g)'(5)=f'(5)g(5)+f(5)g'(5)$. Substitute $f(5)=5$, $f'(5)= - 2$, $g(5)=-7$, and $g'(5)=2$ into the formula: $(f\cdot g)'(5)=(-2)\times(-7)+5\times2=14 + 10=24$

Step5: Use quotient - rule of derivatives

The quotient - rule states that $\left(\frac{f}{g}\right)'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{g^{2}(x)}$. So, $\left(\frac{f}{g}\right)'(5)=\frac{f'(5)g(5)-f(5)g'(5)}{g^{2}(5)}$. Substitute $f(5)=5$, $f'(5)= - 2$, $g(5)=-7$, and $g'(5)=2$ into the formula: $g^{2}(5)=(-7)^{2}=49$, $f'(5)g(5)-f(5)g'(5)=(-2)\times(-7)-5\times2=14 - 10 = 4$. So, $\left(\frac{f}{g}\right)'(5)=\frac{4}{49}$

Answer:

a. $0$ b. $-4$ c. $24$ d. $\frac{4}{49}$