15. If $$ f^{\prime}(x)=\frac{(x+3)-(x+1)}{(x+3)^{2}} $$ is a derivative of a function obtained by using the Quotient Rule, determine the original function $$ f(x)-(2 $$ points) 16. Let $$ h(x) $$ be a function defined for all $$ x
eq 0 $$ such that $$ h(4)=-3 $$ and the derivative of $$ h $$ is given by

Question

15. If $$ f^{\prime}(x)=\frac{(x+3)-(x+1)}{(x+3)^{2}} $$ is a derivative of a function obtained by using the Quotient Rule, determine the original function $$ f(x)-(2 $$ points) 16. Let $$ h(x) $$ be a function defined for all $$ x 
eq 0 $$ such that $$ h(4)=-3 $$ and the derivative of $$ h $$ is given by

UpStudy AI · Accepted Answer

Problem 15:

Given the derivative
$$
f^{\prime}(x) = \frac{(x+3) - (x+1)}{(x+3)^2} = \frac{2}{(x+3)^2},
$$
we want to find the original function $$ f(x) $$.

To find $$ f(x) $$, we'll integrate $$ f^{\prime}(x) $$:

$$
f(x) = \int f^{\prime}(x) \, dx = \int \frac{2}{(x+3)^2} \, dx.
$$

Let's perform the integration:

$$
\int \frac{2}{(x+3)^2} \, dx = 2 \int (x+3)^{-2} \, dx = 2 \left( \frac{(x+3)^{-1}}{-1} \right) + C = -\frac{2}{x+3} + C,
$$
where $$ C $$ is the constant of integration.

**Original Function:**
$$
f(x) = -\frac{2}{x + 3} + C
$$
The original function is $$ f(x) = -\frac{2}{x + 3} + C $$.

If is a derivative of a function obtained by using the Quotient Rule, determine the
original function points)

Let be a function defined for all such that and the derivative of is given by

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If is a derivative of a function obtained by using the Quotient Rule, determine the original function points) Let be a function defined for all such that and the derivative of is given by