Derivative of cos(tan(x))

The solution

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cos(tan(x))

$$\cos{\left(\tan{\left(x \right)} \right)}$$

cos(tan(x))

Detail solution

Let $u = \tan{\left(x \right)}$ .
The derivative of cosine is negative sine:

$\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result of the chain rule is:

$- \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(\tan{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}}$
Now simplify:

$- \frac{\sin{\left(\tan{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}}$

The answer is:

$- \frac{\sin{\left(\tan{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 /       2   \            
-\1 + tan (x)/*sin(tan(x))

$$- \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(\tan{\left(x \right)} \right)}$$

Simplify

The second derivative [src]

 /       2   \ //       2   \                                   \
-\1 + tan (x)/*\\1 + tan (x)/*cos(tan(x)) + 2*sin(tan(x))*tan(x)/

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

Simplify

The third derivative [src]

              /             2                                                                                                       \
/       2   \ |/       2   \                     2                    /       2   \                 /       2   \                   |
\1 + tan (x)/*\\1 + tan (x)/ *sin(tan(x)) - 4*tan (x)*sin(tan(x)) - 2*\1 + tan (x)/*sin(tan(x)) - 6*\1 + tan (x)/*cos(tan(x))*tan(x)/

$$\left(\tan^{2}{\left(x \right)} + 1\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(\tan{\left(x \right)} \right)} - 2 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(\tan{\left(x \right)} \right)} - 6 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)} - 4 \sin{\left(\tan{\left(x \right)} \right)} \tan^{2}{\left(x \right)}\right)$$

Simplify

The graph

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Derivative of cos(tan(x))

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The solution