Consider the following statements Statement 1 : If \(y=\log _{10} x+\log _{c} x\), then \(\frac{d y}{d x}=\frac{\log _{10} e}{x}+\frac{1}{x}\)
Statement 2 : If \(\frac{d}{d x}\left(\log _{10} x\right)=\frac{\log x}{\log 10}\) and \(\frac{d}{d x}\left(\log _{e} x\right)=\frac{\log x}{\log e}\)

Statement 1 ,
\(\begin{aligned}
y &=\log _{10} x+\log _{e} x \\
\Rightarrow \quad y &=\frac{\log _{e} x}{\log _{e} 10}+\log _{e} x
\end{aligned}\)
On differentiating w.r.t. \(x\), we get
\(\begin{aligned} \frac{d y}{d x} &=\frac{1}{x \log _{e} 10}+\frac{1}{x} \\ \Rightarrow \quad \frac{d y}{d x} &=\frac{\log _{10} e}{x}+\frac{1}{x} \\ \text { Statement 2, } \\ \frac{d}{d x} \log _{10} x &=\frac{d}{d x} \frac{\log _{e} x}{\log _{e} 10} \\ &=\frac{\log _{x}}{x \log _{e} 10} \\ \frac{d}{d x}\left(\log _{e} x\right) &=\frac{d}{d x} \frac{\log _{e} x}{\log _{e} e}=\frac{1}{x} \end{aligned}\)
Hence, statement 1 is true but statement 2 is false.