What is the derivative of $f (x) = ln (cos x))$ $f(x) = ln(cosx))$ ?

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Answer 1 · 2018-03-21T15:09:39

$\frac{d f}{d x} = - tan x$ $(df)/(dx)=-tanx$

We have $f (x) = ln (cos x)$ $f(x)=ln(cosx)$ , let $g (x) = cos x$ $g(x)=cosx$ , then $f (x) = ln (g (x))$ $f(x)=ln(g(x))$

Now using chain formula $\frac{d f}{d x} = \frac{d f}{d g (x)} \cdot \frac{d g (x)}{d x}$ $(df)/(dx)=(df)/(dg(x))*(dg(x))/dx$

= $\frac{1}{g (x)} \cdot (- sin x)$ $1/(g(x))*(-sinx)$

= $\frac{1}{cos x} \cdot (- sin x)$ $1/cosx*(-sinx)$

= $- tan x$ $-tanx$

What is the derivative of f(x) = ln(cosx))f(x)=ln(cosx))f(x) = ln(cosx))?