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1/cosx^3
The derivative of the function/ 1/cosx^3

Derivative of 1/cosx^3

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

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The solution

You have entered [src] 
     1   
1*-------
     3   
  cos (x)
1⋅1cos⁡3(x)1 \cdot \frac{1}{\cos^{3}{\left(x \right)}} 
d /     1   \
--|1*-------|
dx|     3   |
  \  cos (x)/
ddx1⋅1cos⁡3(x)\frac{d}{d x} 1 \cdot \frac{1}{\cos^{3}{\left(x \right)}}
Transform
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=1f{\left(x \right)} = 1 and g(x)=cos⁡3(x)g{\left(x \right)} = \cos^{3}{\left(x \right)}.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. The derivative of the constant 11 is zero.

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

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

    2. Apply the power rule: u3u^{3} goes to 3u23 u^{2}

    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:

      −3sin⁡(x)cos⁡2(x)- 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}

    Now plug in to the quotient rule:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-50000005000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   3*sin(x)   
--------------
          3   
cos(x)*cos (x)
3sin⁡(x)cos⁡(x)cos⁡3(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)
3⋅(4sin⁡2(x)cos⁡2(x)+1)cos⁡3(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)
3⋅(20sin⁡2(x)cos⁡2(x)+11)sin⁡(x)cos⁡4(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
Derivative of 1/cosx^3
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