Derivative of ln(sec(x))

The calculator will find the derivative of

\ln\left(\sec{\left(x \right)}\right)

, with steps shown.

Your Input

Find $\frac{d}{dx} \left(\ln\left(\sec{\left(x \right)}\right)\right)$ .

Solution

The function $\ln\left(\sec{\left(x \right)}\right)$ is the composition $f{\left(g{\left(x \right)} \right)}$ of two functions $f{\left(u \right)} = \ln\left(u\right)$ and $g{\left(x \right)} = \sec{\left(x \right)}$ .

Apply the chain rule $\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$ :

{\color{red}\left(\frac{d}{dx} \left(\ln\left(\sec{\left(x \right)}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(\sec{\left(x \right)}\right)\right)}

The derivative of the natural logarithm is $\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$ :

{\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(\sec{\left(x \right)}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(\sec{\left(x \right)}\right)

Return to the old variable:

\frac{\frac{d}{dx} \left(\sec{\left(x \right)}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(\sec{\left(x \right)}\right)}{{\color{red}\left(\sec{\left(x \right)}\right)}}

The derivative of the secant is $\frac{d}{dx} \left(\sec{\left(x \right)}\right) = \tan{\left(x \right)} \sec{\left(x \right)}$ :

\frac{{\color{red}\left(\frac{d}{dx} \left(\sec{\left(x \right)}\right)\right)}}{\sec{\left(x \right)}} = \frac{{\color{red}\left(\tan{\left(x \right)} \sec{\left(x \right)}\right)}}{\sec{\left(x \right)}}

Thus, $\frac{d}{dx} \left(\ln\left(\sec{\left(x \right)}\right)\right) = \tan{\left(x \right)}$ .

Answer

$\frac{d}{dx} \left(\ln\left(\sec{\left(x \right)}\right)\right) = \tan{\left(x \right)}$ A

Derivative of ln⁡(sec⁡(x))\ln\left(\sec{\left(x \right)}\right)ln(sec(x))

Your Input

Solution

Answer

Derivative of $\ln\left(\sec{\left(x \right)}\right)$