Derivative of y=lncos100x

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log(cos(100*x))

$$\log{\left(\cos{\left(100 x \right)} \right)}$$

log(cos(100*x))

Detail solution

Let $u = \cos{\left(100 x \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(100 x \right)}$ :
1. Let $u = 100 x$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 100 x$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $100$
  The result of the chain rule is:
  
  $- 100 \sin{\left(100 x \right)}$
The result of the chain rule is:

$- \frac{100 \sin{\left(100 x \right)}}{\cos{\left(100 x \right)}}$
Now simplify:

$- 100 \tan{\left(100 x \right)}$

The answer is:

$- 100 \tan{\left(100 x \right)}$

The graph

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The first derivative [src]

-100*sin(100*x)
---------------
   cos(100*x)

$$- \frac{100 \sin{\left(100 x \right)}}{\cos{\left(100 x \right)}}$$

Simplify

The second derivative [src]

       /       2       \
       |    sin (100*x)|
-10000*|1 + -----------|
       |       2       |
       \    cos (100*x)/

$$- 10000 \left(\frac{\sin^{2}{\left(100 x \right)}}{\cos^{2}{\left(100 x \right)}} + 1\right)$$

Simplify

The third derivative [src]

         /       2       \           
         |    sin (100*x)|           
-2000000*|1 + -----------|*sin(100*x)
         |       2       |           
         \    cos (100*x)/           
-------------------------------------
              cos(100*x)

$$- \frac{2000000 \left(\frac{\sin^{2}{\left(100 x \right)}}{\cos^{2}{\left(100 x \right)}} + 1\right) \sin{\left(100 x \right)}}{\cos{\left(100 x \right)}}$$

Simplify

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Derivative of y=lncos100x

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