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The derivative of the function/ 1/cosx^3

Derivative of 1/cosx^3

The solution

You have entered [src]

     1   
1*-------
     3   
  cos (x)

$$1 \cdot \frac{1}{\cos^{3}{\left(x \right)}}$$

d /     1   \
--|1*-------|
dx|     3   |
  \  cos (x)/

$$\frac{d}{d x} 1 \cdot \frac{1}{\cos^{3}{\left(x \right)}}$$

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = 1$ and $g{\left(x \right)} = \cos^{3}{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of the constant $1$ is zero.
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \cos{\left(x \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. 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:
  
  $- 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}$
Now plug in to the quotient rule:

$\frac{3 \sin{\left(x \right)}}{\cos^{4}{\left(x \right)}}$

The answer is:

$\frac{3 \sin{\left(x \right)}}{\cos^{4}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   3*sin(x)   
--------------
          3   
cos(x)*cos (x)

$$\frac{3 \sin{\left(x \right)}}{\cos{\left(x \right)} \cos^{3}{\left(x \right)}}$$

Simplify

The second derivative [src]

  /         2   \
  |    4*sin (x)|
3*|1 + ---------|
  |        2    |
  \     cos (x) /
-----------------
        3        
     cos (x)

$$\frac{3 \cdot \left(\frac{4 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right)}{\cos^{3}{\left(x \right)}}$$

Simplify

The third derivative [src]

  /           2   \       
  |     20*sin (x)|       
3*|11 + ----------|*sin(x)
  |         2     |       
  \      cos (x)  /       
--------------------------
            4             
         cos (x)

$$\frac{3 \cdot \left(\frac{20 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 11\right) \sin{\left(x \right)}}{\cos^{4}{\left(x \right)}}$$

Simplify

The graph

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Identical expressions

Derivative of 1/cosx^3

The graph:

Piecewise:

The solution