If $x^{m}y^{n}=(x+y)^{m+n}$, prove that $\dfrac{d^{2}y}{dx^{2}}=0$.

CBSE Class 12 · 2017 · DifferentiationLong Answer

x^{m} y^{n} = (x + y)^{m + n}

, prove that

\frac{d ^{2} y}{d x ^{2}} = 0

Answer: Proved

Take logarithm:

m lo g x + n lo g y = (m + n) lo g (x + y)

Differentiate w.r.t.

x

\frac{m}{x} + \frac{n}{y} \frac{d y}{d x} = \frac{m + n}{x + y} (1 + \frac{d y}{d x})

Simplifying (cross-multiplying and cancelling) leads to the clean first derivative:

\frac{d y}{d x} = \frac{y}{x}

Differentiate again by the quotient rule and substitute

\frac{d y}{d x} = y / x

\frac{d ^{2} y}{d x ^{2}} = \frac{x \frac{d y}{d x} - y}{x ^{2}} = \frac{x \cdot \frac{y}{x} - y}{x ^{2}} = 0

\frac{d ^{2} y}{d x ^{2}} = 0

CBSE Class 12 Board · 2017 · Subjective