#### Provide solution for RD Sharma maths class 12 chapter Higher Order Derivatives exercise 11.1 question 1 subquestion (vi)

$5x+6x\: log\: x$

HInt:

You must know about derivative of log x & x3

Given:

$x^{3}\: log\: x$

Solution:

$Let\: \: y=x^{3}\: log\: x$

Use multiplicative rule

As UV=UV1+U1V

Where U=x3 & V=log x

\begin{aligned} &\frac{d y}{d x}=x^{3} \frac{d}{d x} \log x+\frac{d}{d x} x^{3} \log x \\ &\frac{d y}{d x}=x^{3} \frac{1}{x}+3 x^{2} \log x \quad\left(\frac{d}{d x} \log x=\frac{1}{x}, \frac{d}{d x} x^{3}=3 x^{2}\right) \\ &\ \end{aligned}

Use multiplicative rule

As UV=UV1+U1V

Where U=x2 & V=log x

\begin{aligned} &\frac{d^{2} y}{d x^{2}}=\frac{d}{d x} x^{2}+3\left[x^{2} \frac{d}{d x} \log x+\frac{d}{d x} x^{2} \cdot \log x\right] \\ &\left.\frac{d^{2} y}{d x^{2}}=2 x+3[x+2 x \cdot \log x] \quad \text { ( } \frac{d}{d x} x^{2}=2 x, \frac{d}{d x} \log x=\frac{1}{x}\right) \\ &\frac{d^{2} y}{d x^{2}}=2 x+3 x+6 x \cdot \log x \\ &\frac{d^{2} y}{d x^{2}}=5 x+6 x \log x \end{aligned}