Question

Find the second order derivative of y=3x^2 1 + log⁡(4x)

(a) 3+\(\frac{1}{x^2}\)

(b) 3-\(\frac{1}{x^2}\)

(c) 6-\(\frac{1}{x^2}\)

(d) 6+\(\frac{1}{x^2}\)

This question was addressed to me in an internship interview.

I would like to ask this question from Second Order Derivatives in portion Continuity and Differentiability of Mathematics – Class 12

Answer 1

Correct choice is (c) 6-\(\frac{1}{x^2}\)

The best I can explain: Given that, y=3x^2+log⁡(4x)

\(\frac{dy}{dx}=6x+\frac{1}{4x}.4=6x+\frac{1}{x}=\frac{6x^2+1}{x}\)

\(\frac{d^2 y}{dx^2}=\frac{\frac{d}{dx} (6x^2+1).(x)-\frac{d}{dx} (x).(6x^2+1)}{x^2} \Big(using\, \frac{d}{dx} (\frac{u}{v})=\frac{(\frac{d}{dx} (u).v-\frac{d}{dx} (v).u)}{v^2}\Big)\)

\(\frac{d^2 y}{dx^2}=\frac{(12x.x-6x^2-1)}{x^2} \)

\(\frac{d^2 y}{dx^2}=\frac{6x^2-1}{x^2} = 6-\frac{1}{x^2}\).