Question

If y = x³ log x, prove that \[\frac{d^4 y}{d x^4} = \frac{6}{x}\] ?

Answer 1

Here,

\[y = x^3 \log x\]
\[\text { Differentiating w . r . t . x, we get }\]
\[\frac{d y}{d x} = 3 x^2 \log x + x^3 \times \frac{1}{x}\]
\[ = 3 x^2 \log x + x^2 \]
\[\text { Differentiating again w . r . t . x, we get }\]
\[\frac{d^2 y}{d x^2} = 6x \log x + 3 x^2 \times \frac{1}{x} + 2x \]
\[ = 6x \log x + 5x\]
\[\text { Differentiating again w . r . t . x, we get }\]
\[\frac{d^3 y}{d x^3} = 6\log x + 6x \times \frac{1}{x} + 5 = 6 \log x + 11\]
\[\text { Differentiating again w . r . t . x, we get }\]
\[\frac{d^4 y}{d x^4} = \frac{6}{x}\]

Hence proved.