KCET · Maths · Application of Derivatives

If \(y=2 x^{n+1}+\frac{3}{x^{n}}\), then \(x^{2} \frac{d^{2 y}}{d x^{2}}\) is

A \(6 n(n+1) y\)
B \(n(n+1) y\)
C \(x \frac{d y}{d x}+y\)
D \(y\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(n(n+1) y\)

Step-by-step Solution

Detailed explanation

We have, \(y\)
\(\begin{aligned}
y &=2 x^{n+1}+\frac{3}{x^{n}} \\
&=2 x^{n+1}+3 x^{-n}...(i)
\end{aligned}\)
On differentiating Eq. (i) both sides w.r.t. \(x\),
we get
\(\frac{d y}{d x}=2(n+1) x^{n}+3(-n) x^{-n-1}\)
Again differentiate w.r.t. to \(x\), then we get
\(\frac{d^{2} y}{d x^{2}}=2(n+1)(n) x^{n-1}+3(-n)(-n-1) x^{-n-2}\)
\(=n(n+1)\left(2 x^{n-1}+3 x^{-n-2}\right)\)
\(\Rightarrow \quad x^{2} \frac{d^{2} y}{d x^{2}}=n(n+1)\left(2 x^{n+1}+\frac{3}{x^{n}}\right)\)
\(\Rightarrow \quad x^{2} \frac{d y}{d x}=n(n+1) y\) [using Eq. (i)]

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Match List-I with List-II

List-I		List-II
a)	A matrix which is not a square matrix	i)	Symmetric matrix
b)	A square matrix \(A = A'\)	ii)	Null matrix
c)	The diagonal elements of a diagonal matrix are same	iii)	Rectangular matrix
d)	A matrix which is both symmetric and skew symmetric	iv)	Scalar matrix

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