Question

The derivative of log₁₀x with respect to x is ___________________.

Answer 1

Let $y={\log }_{10}x$.

$y={\log }_{10}x$

$\Rightarrow y=\frac{{\log }_{e}x}{{\log }_{e}10}\left({\log }_{b}a=\frac{{\log }_{c}a}{{\log }_{c}b}\right)$

Differentiating both sides with respect to x, we get

$\frac{dy}{dx}=\frac{d}{dx}\left(\frac{{\log }_{e}x}{{\log }_{e}10}\right)$

$\Rightarrow \frac{dy}{dx}=\frac{1}{{\log }_{e}10}\times \frac{d}{dx}\left({\log }_{e}x\right)$

$\Rightarrow \frac{dy}{dx}=\frac{1}{{\log }_{e}10}\times \frac{1}{x}$

$\Rightarrow \frac{dy}{dx}=\frac{1}{x{\log }_{e}10}$

Thus, the derivative of ${\log }_{10}x$ with respect to x is $\frac{1}{x{\log }_{e}10}$.

The derivative of log₁₀x with respect to x is $\underline{\frac{1}{x{\log }_{e}10}}$.