Q.If $\log_5\dfrac{x^4+y^4}{x^4-y^4}=2$, show $\dfrac{dy}{dx}=-\dfrac{12x^3}{13y^3}$.

Given $\log_5\dfrac{x^4+y^4}{x^4-y^4}=2$.

Step 1 — undo the log: $\log_5(\cdot)=2\implies(\cdot)=5^2=25$.

Step 2 — collect like terms to see the actual curve:

(check: at $x=1$, $y^4=12/13\approx0.923$, and $(1+0.923)/(1-0.923)\approx25$ ✓.)

Step 3 — differentiate $x^4+y^4=25x^4-25y^4$ implicitly:

Step 4 — collect $dy/dx$ terms (both coefficients land on the same side with the same sign, since $26y^4=24x^4$ has $x^4$ and $y^4$ varying together, not oppositely):

$104y^3\frac{dy}{dx}=96x^3$ …

Undo the base-5 log to get an algebraic relation, differentiate implici …

Assuming the answer must be negative by analogy with similar-looking problems (like the difference-over-sum form) — here $x^4$ and $y^4$ actually increase together o …