Animated Solution for Mathematics - Differentiation: If $(a+\sqrt{2}b\cos x)(a-\sqrt{2}b\cos y)=a^2-b^2$, where $a>b>0$, then $\frac{dx}{dy}$ at $\left(\frac{\pi }{4},\frac{\pi }{4}\right)$ is:

We are observing a delicate dance between two variables, $x$ and $y$. We are given the relationship:

Here, we are constrained by $a>b>0$. Our mission is to find the rate of change of $x$ with respect to $y$, denoted as $\frac{dx}{dy}$, at the specific coordinate $(\frac{\pi }{4},\frac{\pi }{4})$.

When you look at this equation, your instinct might be to expand it. Resist that urge! In the world of calculus, elegance often lies in keeping structure intact.

We treat this as a product of two functions, $u$ and $v$, where $u=(a+\sqrt{2}b\cos x)$ and $v=(a-\sqrt{2}b\cos y)$. To find $\frac{dx}{dy}$, we apply the Product Rule:

Note that the derivative of the right side, $a^2-b^2$, is zero because $a$ and $b$ are constants. This is our first victory—the right side vanishes, simplifying our life significantly.

Now, let us differentiate each part. For $u$, we must be careful. Since $u$ is a function of $x$, and $x$ is a function of $y$, we use the Chain Rule:

For $v$, it is simpler because it is already a function of $y$:

Now, we assemble our pieces into the Product Rule formula:

At $x=\frac{\pi }{4}$ and $y=\frac{\pi }{4}$, we know that $\sin (\frac{\pi }{4})=\cos (\frac{\pi }{4})=\frac{1}{\sqrt{2}}$. Watch what happens when we substitute these values.

The $\sqrt{2}$ terms, which seemed like a nuisance, now cancel out perfectly with the $\frac{1}{\sqrt{2}}$ factors. The equation collapses into:

Since $b>0$, we can divide the entire equation by $b$ without a second thought. We are left with:

With a simple rearrangement, we isolate our target:

Dividing by $(a-b)$, we arrive at the elegant result:

Look at that result. It is clean, symmetric, and perfectly derived. You have successfully navigated the implicit landscape. The final answer is $\frac{a+b}{a-b}$.