Animated Solution for Mathematics - Differentiation: Let $y={\log }_e(\frac{1-x^2}{1+x^2}),-1<x<1$. Then at $x=\frac{1}{2}$, the value of $225(y^{\prime }-y^{\prime \prime })$ is equal to

Imagine standing before a massive, intimidating mountain. If you try to climb it straight up the sheer face, you will likely fall. But if you look for the winding path, the climb becomes a journey.

We are given the function:

If you jump straight into the quotient rule, you are climbing the sheer face. Instead, let's use the power of logarithmic properties. We know that ${\log }_e(\frac{a}{b})={\log }_e(a)-{\log }_e(b)$.

This simple identity transforms our function into:

Now, the mountain is just a gentle hill.

Now that we have simplified, let's find $y^{\prime }$. We apply the chain rule to each term.

The derivative of ${\log }_e(1-x^2)$ is $\frac{-2x}{1-x^2}$, and the derivative of ${\log }_e(1+x^2)$ is $\frac{2x}{1+x^2}$. When we combine these, we get:

Factoring out $-2x$, we find a common denominator of $(1-x^2)(1+x^2)$, which is $1-x^4$. The numerator simplifies beautifully to $(1+x^2)+(1-x^2)=2$.

Thus, the first derivative is:

Now, we face the second derivative, $y^{\prime \prime }$. We must differentiate $\frac{-4x}{1-x^4}$ using the quotient rule: $\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$.

Here, $u=-4x$ and $v=1-x^4$. The derivatives are $u^{\prime }=-4$ and $v^{\prime }=-4x^3$. Plugging these in, we get:

Simplifying the numerator, we distribute the $-4$ to get $-4+4x^4$, and the second term becomes $-16x^4$. Combining these, we get $-4-12x^4$, or $-4(1+3x^4)$.

So, the second derivative is:

We are at the finish line. We need to evaluate $225(y^{\prime }-y^{\prime \prime })$ at $x=\frac{1}{2}$.

First, let's find the value of $x^4=(\frac{1}{2}{)}^4=\frac{1}{16}$. Then $1-x^4=1-\frac{1}{16}=\frac{15}{16}$.

Now, substitute these into our derivatives: For $y^{\prime }$, we get:

For $y^{\prime \prime }$, we get:

Now, compute $y^{\prime }-y^{\prime \prime }$:

Converting to a common denominator of $225$, we get:

Finally, multiply by $225$:

The cancellation is perfect. The final answer is 736.