Question

question: for (j(x)=\frac{f(x)}{g(x)}), use the quotient - rule to find (j(-1)) if (f(-1)=-4), (f(-1)=3), (g(-1)=1), and (g(-1)=-3). enter an exact answer. provide your answer below: (j(-1)=square)

Explanation:

Step1: Recall the quotient - rule formula

The quotient - rule states that if $j(x)=\frac{f(x)}{g(x)}$, then $j^{\prime}(x)=\frac{f^{\prime}(x)g(x)-f(x)g^{\prime}(x)}{[g(x)]^{2}}$.

Step2: Substitute $x = - 1$ into the quotient - rule formula

We are given that $f(-1)=-4$, $f^{\prime}(-1)=3$, $g(-1)=1$, and $g^{\prime}(-1)=-3$.
Substituting these values into the quotient - rule formula $j^{\prime}(-1)=\frac{f^{\prime}(-1)g(-1)-f(-1)g^{\prime}(-1)}{[g(-1)]^{2}}$.

Step3: Calculate the numerator

$f^{\prime}(-1)g(-1)-f(-1)g^{\prime}(-1)=(3\times1)-(-4\times(-3))=3 - 12=-9$.

Step4: Calculate the denominator

$[g(-1)]^{2}=1^{2}=1$.

Step5: Calculate $j^{\prime}(-1)$

$j^{\prime}(-1)=\frac{-9}{1}=-9$.

Answer:

$-9$