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Derivative of:
Derivative of lnx^6
Derivative of y=ln^3(sec(x))
Derivative of 3cosx-4sinx
Derivative of sqrt(4-5x)
Identical expressions
y=ln^ three (sec(x))
y equally ln cubed (sec(x))
y equally ln to the power of three (sec(x))
y=ln3(sec(x))
y=ln3secx
y=ln³(sec(x))
y=ln to the power of 3(sec(x))
y=ln^3secx

The derivative of the function/ y=ln^3(sec(x))

Derivative of y=ln^3(sec(x))

The solution

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   3        
log (sec(x))

$$\log{\left(\sec{\left(x \right)} \right)}^{3}$$

d /   3        \
--\log (sec(x))/
dx

$$\frac{d}{d x} \log{\left(\sec{\left(x \right)} \right)}^{3}$$

Transform

Detail solution

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

$\frac{3 \log{\left(\sec{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)} \sec{\left(x \right)}}$
Now simplify:

$3 \log{\left(\frac{1}{\cos{\left(x \right)}} \right)}^{2} \tan{\left(x \right)}$

The answer is:

$3 \log{\left(\frac{1}{\cos{\left(x \right)}} \right)}^{2} \tan{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     2               
3*log (sec(x))*tan(x)

$$3 \log{\left(\sec{\left(x \right)} \right)}^{2} \tan{\left(x \right)}$$

Simplify

The second derivative [src]

  /     2      /       2   \            \            
3*\2*tan (x) + \1 + tan (x)/*log(sec(x))/*log(sec(x))

$$3 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\sec{\left(x \right)} \right)} + 2 \tan^{2}{\left(x \right)}\right) \log{\left(\sec{\left(x \right)} \right)}$$

Simplify

The third derivative [src]

  /   2         2         /       2   \     /       2   \            \       
6*\tan (x) + log (sec(x))*\1 + tan (x)/ + 3*\1 + tan (x)/*log(sec(x))/*tan(x)

$$6 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\sec{\left(x \right)} \right)}^{2} + 3 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\sec{\left(x \right)} \right)} + \tan^{2}{\left(x \right)}\right) \tan{\left(x \right)}$$

Simplify

The graph

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Identical expressions

Derivative of y=ln^3(sec(x))

The graph:

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The solution