Let $$ f(x)=\frac{\sqrt{x}-5}{\sqrt{x}+5} $$. Find $$ f^{\prime}(x) $$ $$ f^{\prime}(x)=\square $$ Find $$ f^{\prime}(25) $$ $$ f^{\prime}(25)=\square $$

Question

UpStudy AI · Accepted Answer

To find the derivative $$ f'(x) $$ of the function $$ f(x) = \frac{\sqrt{x} - 5}{\sqrt{x} + 5} $$, we will use the quotient rule. The quotient rule states that if $$ f(x) = \frac{g(x)}{h(x)} $$, then

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

In our case, let $$ g(x) = \sqrt{x} - 5 $$ and $$ h(x) = \sqrt{x} + 5 $$.

First, we need to find $$ g'(x) $$ and $$ h'(x) $$:

1. **Finding $$ g'(x) $$**:
   $$
   g(x) = \sqrt{x} - 5 \implies g'(x) = \frac{1}{2\sqrt{x}}
   $$

2. **Finding $$ h'(x) $$**:
   $$
   h(x) = \sqrt{x} + 5 \implies h'(x) = \frac{1}{2\sqrt{x}}
   $$

Now we can apply the quotient rule:

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

Substituting $$ g(x) $$, $$ g'(x) $$, $$ h(x) $$, and $$ h'(x) $$:

$$
f'(x) = \frac{\left(\frac{1}{2\sqrt{x}}\right)(\sqrt{x} + 5) - (\sqrt{x} - 5)\left(\frac{1}{2\sqrt{x}}\right)}{(\sqrt{x} + 5)^2}
$$

Simplifying the numerator:

$$
= \frac{\frac{1}{2\sqrt{x}}(\sqrt{x} + 5) - \frac{1}{2\sqrt{x}}(\sqrt{x} - 5)}{(\sqrt{x} + 5)^2}
$$

Combining the terms in the numerator:

$$
= \frac{\frac{1}{2\sqrt{x}} \left( (\sqrt{x} + 5) - (\sqrt{x} - 5) \right)}{(\sqrt{x} + 5)^2}
$$

$$
= \frac{\frac{1}{2\sqrt{x}} \left( \sqrt{x} + 5 - \sqrt{x} + 5 \right)}{(\sqrt{x} + 5)^2}
$$

$$
= \frac{\frac{1}{2\sqrt{x}} (10)}{(\sqrt{x} + 5)^2}
$$

$$
= \frac{10}{2\sqrt{x}(\sqrt{x} + 5)^2}
$$

$$
= \frac{5}{\sqrt{x}(\sqrt{x} + 5)^2}
$$

Thus, we have:

$$
f'(x) = \frac{5}{\sqrt{x}(\sqrt{x} + 5)^2}
$$

Next, we need to find $$ f'(25) $$:

Substituting $$ x = 25 $$:

$$
f'(25) = \frac{5}{\sqrt{25}(\sqrt{25} + 5)^2}
$$

Calculating $$ \sqrt{25} = 5 $$:

$$
f'(25) = \frac{5}{5(5 + 5)^2} = \frac{5}{5(10)^2} = \frac{5}{5 \cdot 100} = \frac{5}{500} = \frac{1}{100}
$$

Thus, the final answers are:

$$
f'(x) = \frac{5}{\sqrt{x}(\sqrt{x} + 5)^2}
$$

$$
f'(25) = \frac{1}{100}
$$
$$ f'(x) = \frac{5}{\sqrt{x}(\sqrt{x} + 5)^2} $$ $$ f'(25) = \frac{1}{100} $$

Let . Find

Find

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Calculus Questions

Let . Find Find