Question

Differentiate the function.

14. \(y = {\log _{10}}\sec x\)

Short answer

The derivative of the function is \(\left( {\frac{{\tan x}}{{\log 10}}} \right)\).

Step-by-step solution

The derivative of the function is \(\left( {\frac{{\tan x}}{{\log 10}}} \right)\).

Rule 2: The derivative of \(\ln x\) is,

\(\frac{d}{{dx}}\left( {\ln x} \right) = \frac{1}{x}\)

Evaluating the derivative of given function

\(\begin{aligned}{c}y&= {\log _{10}}\left( {\sec x} \right)\\y&= \frac{{\log \sec x}}{{\log 10}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{{\log }_b} = \frac{{\log a}}{{\log b}}} \right)\end{aligned}\)

Differentiating y with respect to x,

\(\begin{aligned}{l}\frac{{dy}}{{dx}}&= \frac{1}{{\log 10}} \times \frac{{d\log \sec x}}{{dx}}\\\frac{{dy}}{{dx}}&= \frac{1}{{\log 10}} \times \frac{{d\log \sec x}}{{d\sec x}} \times \frac{{d\sec x}}{{dx}}\\\frac{{dy}}{{dx}}&= \frac{1}{{\log 10}} \times \frac{1}{{\sec x}} \times \left( {\sec x} \right)\left( {\tan x} \right)\\\frac{{dy}}{{dx}}&= \left( {\frac{{\tan x}}{{\log 10}}} \right)\end{aligned}\)

Thus, the value of the derivative is \(\left( {\frac{{\tan x}}{{\log 10}}} \right)\).