Question

y=(log x)^3 dy/dx= Don't give absurd answers to like 1) Check in google

Answer 1

Chain rule states that,

$\displaystyle\longrightarrow\sf{y=f(g(x))}$

$\displaystyle\Longrightarrow\sf{\dfrac {dy}{dx}=\dfrac {d\,[f(g(x))]}{d\,[g(x)]}\cdot\dfrac {d\,[g(x)]}{dx}}$

In the question,

$\displaystyle\longrightarrow\sf{y=(\log x)^3\quad\quad\dots (1)}$

Let,

$\displaystyle\longrightarrow\sf{u=\log x}$

$\displaystyle\longrightarrow\sf{\dfrac {du}{dx}=\dfrac {1}{x}}$

Then (1) becomes,

$\displaystyle\longrightarrow\sf{y=u^3}$

Differentiating wrt $\displaystyle\sf {x,}$

$\displaystyle\longrightarrow\sf{\dfrac {dy}{dx}=\dfrac {d\,\left[u^3\right]}{dx}}$

By chain rule,

$\displaystyle\longrightarrow\sf{\dfrac {dy}{dx}=\dfrac {d\,\left[u^3\right]}{du}\cdot\dfrac {du}{dx}}$

$\displaystyle\longrightarrow\sf{\dfrac {dy}{dx}=3u^2\cdot\dfrac {du}{dx}}$

$\displaystyle\longrightarrow\sf{\dfrac {dy}{dx}=3(\log x)^2\cdot\dfrac {1}{x}}$

$\displaystyle\longrightarrow\underline {\underline {\sf{\dfrac {dy}{dx}=\dfrac {3(\log x)^2}{x}}}}$