Question

Find the second order derivative of the function $y=\log x$

Answer 1

Finding $\frac{dy}{dx}$
Let $y=\log x$
Differentiating both sides w.r.t. $x$ we get
$\frac{dy}{dx}=\frac{d\left(\log x\right)}{dx}$
$\Rightarrow \frac{dy}{dx}=\frac{1}{x}\cdots \left(i\right)$

Finding $\frac{d^{2}y}{d{x}^2}$
Again, differentiating both sides of $\left(i\right)$ w.r.t. $x$, we get
$\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}\left(\frac{1}{x}\right)$
$\Rightarrow \frac{d^{2}y}{d{x}^2}=\frac{d\left(x^{-1}\right)}{dx}$
$\Rightarrow \frac{d^{2}y}{d{x}^2}=-1.x^{-1-1}$
$\Rightarrow \frac{d^{2}y}{d{x}^2}=-x^{-2}$
$\Rightarrow \frac{d^{2}y}{d{x}^2}=\frac{-1}{x^2}$