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Derivative of:
Derivative of 4/x^2
Derivative of cos(x^3)
Derivative of (cos(x))^3
Derivative of 1/tan(x)
Identical expressions
one /cos^6t
1 divide by co sinus of e of to the power of 6t
one divide by co sinus of e of to the power of 6t
1/cos6t
1/cos⁶t
1 divide by cos^6t

The derivative of the function/ 1/cos^6t

Derivative of 1/cos^6t

The solution

You have entered [src]

   1   
-------
   6   
cos (t)

$$\frac{1}{\cos^{6}{\left(t \right)}}$$

1/(cos(t)^6)

Detail solution

Let $u = \cos^{6}{\left(t \right)}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d t} \cos^{6}{\left(t \right)}$ :
1. Let $u = \cos{\left(t \right)}$ .
2. Apply the power rule: $u^{6}$ goes to $6 u^{5}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d t} \cos{\left(t \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)}$
  The result of the chain rule is:
  
  $- 6 \sin{\left(t \right)} \cos^{5}{\left(t \right)}$
The result of the chain rule is:

$\frac{6 \sin{\left(t \right)}}{\cos^{7}{\left(t \right)}}$

The answer is:

$\frac{6 \sin{\left(t \right)}}{\cos^{7}{\left(t \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   6*sin(t)   
--------------
          6   
cos(t)*cos (t)

$$\frac{6 \sin{\left(t \right)}}{\cos{\left(t \right)} \cos^{6}{\left(t \right)}}$$

Simplify

The second derivative [src]

  /         2   \
  |    7*sin (t)|
6*|1 + ---------|
  |        2    |
  \     cos (t) /
-----------------
        6        
     cos (t)

$$\frac{6 \left(\frac{7 \sin^{2}{\left(t \right)}}{\cos^{2}{\left(t \right)}} + 1\right)}{\cos^{6}{\left(t \right)}}$$

Simplify

The third derivative [src]

   /          2   \       
   |    14*sin (t)|       
24*|5 + ----------|*sin(t)
   |        2     |       
   \     cos (t)  /       
--------------------------
            7             
         cos (t)

$$\frac{24 \left(\frac{14 \sin^{2}{\left(t \right)}}{\cos^{2}{\left(t \right)}} + 5\right) \sin{\left(t \right)}}{\cos^{7}{\left(t \right)}}$$

Simplify

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Identical expressions

Derivative of 1/cos^6t

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Piecewise:

The solution