Mister Exam

Other calculators

ln(cos(x))
The derivative of the function/ ln(cos(x))

Derivative of ln(cos(x))

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
log(cos(x))
log⁡(cos⁡(x))\log{\left(\cos{\left(x \right)} \right)} 
d              
--(log(cos(x)))
dx
ddxlog⁡(cos⁡(x))\frac{d}{d x} \log{\left(\cos{\left(x \right)} \right)}
Transform
Detail solution

  1. Let u=cos⁡(x)u = \cos{\left(x \right)}.

  2. The derivative of log⁡(u)\log{\left(u \right)} is 1u\frac{1}{u}.

  3. Then, apply the chain rule. Multiply by ddxcos⁡(x)\frac{d}{d x} \cos{\left(x \right)}:

    1. The derivative of cosine is negative sine:

      ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

    The result of the chain rule is:

    −sin⁡(x)cos⁡(x)- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}

  4. Now simplify:

    −tan⁡(x)- \tan{\left(x \right)}

The answer is:

−tan⁡(x)- \tan{\left(x \right)}

The graph
02468-8-6-4-2-1010-5050
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
-sin(x) 
--------
 cos(x)
−sin⁡(x)cos⁡(x)- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}
Simplify
The second derivative [src] 
 /       2   \
 |    sin (x)|
-|1 + -------|
 |       2   |
 \    cos (x)/
−(sin⁡2(x)cos⁡2(x)+1)- (\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1)
Simplify
The third derivative [src] 
   /       2   \       
   |    sin (x)|       
-2*|1 + -------|*sin(x)
   |       2   |       
   \    cos (x)/       
-----------------------
         cos(x)
−2(sin⁡2(x)cos⁡2(x)+1)sin⁡(x)cos⁡(x)- \frac{2 \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)}}
Simplify
The graph
Derivative of ln(cos(x))
    © Mister Exam – Calculator