Suppose $$ f $$ and $$ g $$ are both defined and differentiable at $$ x=0 $$. Suppose that $$ f(0)=-5, g(0)=1, f^{\prime}(0)=7 $$ and $$ g^{\prime}(0)=3 $$. Then $$ \left(\frac{g}{f}\right)^{\prime}(0)=\frac{A}{B} $$. The value of $$ A $$ is

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We start with the quotient rule, which states that for $$ h(x)=\frac{g(x)}{f(x)} $$ we have

$$
h'(x)=\frac{g'(x)f(x)-g(x)f'(x)}{(f(x))^2}.
$$

Given $$ f(0)=-5 $$, $$ g(0)=1 $$, $$ f'(0)=7 $$, and $$ g'(0)=3 $$, we find

$$
\left(\frac{g}{f}\right)'(0)=\frac{3(-5)-1\cdot7}{(-5)^2}=\frac{-15-7}{25}=\frac{-22}{25}.
$$

Thus, in the expression $$\frac{A}{B}$$, we have $$ A=-22 $$ and $$ B=25 $$.

The value of $$ A $$ is $$-22$$.
The value of $$ A $$ is $$-22$$.

Suppose and are both defined and differentiable at .
Suppose that and .
Then .
The value of is

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Suppose and are both defined and differentiable at . Suppose that and . Then . The value of is