Question

Find ${\displaystyle \frac{\text{d}y}{\text{d}x}}$, if y = (log x)^x + (x)log^x

Answer 1

y = (log x)^x + (x)log^x 

Let u = (log x)^x and v = xlog^x 

∴ y = u + v

Differentiating both sides w. r. t. x, we get

${\displaystyle \frac{\text{d}y}{\text{d}x}=\frac{\text{du}}{\text{d}x}+\frac{\text{dv}}{\text{d}x}}$   .....(i)

Now, u = (log x)^x

Taking logarithm of both sides, we get

log u = log (log x)^x = x log (log x)

Differentiating both sides w. r. t. x, we get

${\displaystyle \frac{\text{d}}{\text{d}x}\left(\log \text{u}\right)=x\cdot \frac{\text{d}}{\text{d}x}\left[\log \left(\log x\right)\right]+\log \left(\log x\right)\cdot \frac{\text{d}}{\text{d}x}\left(x\right)}$

∴ ${\displaystyle \frac{1}{\text{u}}.\frac{\text{du}}{\text{d}x}=x\cdot \frac{1}{\log x}\cdot \frac{\text{d}}{\text{d}x}\left(\log x\right)+\log \left(\log x\right)\cdot 1}$

∴ ${\displaystyle \frac{1}{\text{u}}\cdot \frac{\text{du}}{\text{d}x}=x\cdot \frac{1}{\log x}\cdot \frac{1}{x}+\log \left(\log x\right)}$

∴ ${\displaystyle \frac{\text{du}}{\text{d}x}=\text{u}[\frac{1}{\log x}+\log \left(\log x\right)]}$

∴ ${\displaystyle \frac{\text{du}}{\text{d}x}={\left(\log x\right)}^x[\frac{1}{\log x}+\log \left(\log x\right)]}$  .....(ii)

Also, v = x^logx

Taking logarithm of both sides, we get

log v = log (x^logx) = log x (log x)

∴ log v = (log x)² 

Differentiating both sides w.r.t. x, we get

${\displaystyle \frac{1}{\text{v}}\cdot \frac{\text{dv}}{\text{d}x}=2\log x\cdot \frac{\text{d}}{\text{d}x}\left(\log x\right)}$

∴ ${\displaystyle \frac{1}{\text{v}}\cdot \frac{\text{dv}}{\text{d}x}=2\log x\cdot \frac{1}{x}}$

Substituting (ii) and (iii) in (i), we get∴ ${\displaystyle \frac{\text{d}}{\text{d}x}=\text{v}\left[\frac{2\log x}{x}\right]}$

∴ ${\displaystyle \frac{\text{dv}}{\text{d}x}=x^{\log x}\left[\frac{2\log x}{x}\right]}$   ......(iii)

Substituting (ii) and (iii) in (i), we get

${\displaystyle \frac{\text{d}y}{\text{d}x}={\left(\log x\right)}^x[\frac{1}{\log x}+\log \left(\log x\right)]+x^{\log x}\left[\frac{2\log x}{x}\right]}$