Question

Find the correct values for \(\frac{∂f}{∂x} \,and\, \frac{∂f}{∂y}\) for the function \(f=\frac{2}{x^3}y^2+4y^3.\)

(a) \(\frac{∂f}{∂x}= \frac{-6}{x^2}, \frac{∂f}{∂y}= \frac{2}{x^3}  y+8y^2\)

(b) \(\frac{∂f}{∂x}= \frac{2}{x^4}, \frac{∂f}{∂y}= \frac{2}{x^3}  y+12y^2\)

(c) \(\frac{∂f}{∂x}= \frac{-6}{x^4}, \frac{∂f}{∂y}= \frac{4}{x^3}  y+12y^2\)

(d) \(\frac{∂f}{∂x}= \frac{-6}{x^4}, \frac{∂f}{∂y}= \frac{4}{x^3} y^2+12\)

The question was asked in an interview for job.

Asked question is from Variable Treated as Constant in chapter Partial Differentiation of Engineering Mathematics

Answer 1

The correct choice is (c) \(\frac{∂f}{∂x}= \frac{-6}{x^4}, \frac{∂f}{∂y}= \frac{4}{x^3}  y+12y^2\)

Best explanation: Given: \(f=\frac{2}{x^3}y^2+4y^3\)

Differentiating partially with respect to x we get,

\(\frac{∂f}{∂x}=2\frac{∂(x^{-3})}{∂x}=2(-3x^{-3-1}) = \frac{-6}{x^4}\)

Differentiating with respect to y we get,

\(\frac{∂f}{∂y}= \frac{4}{x^3}y+12y^2\)