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Identical expressions
y=tgx^ three *cosx
y equally tgx cubed multiply by co sinus of e of x
y equally tgx to the power of three multiply by co sinus of e of x
y=tgx3*cosx
y=tgx³*cosx
y=tgx to the power of 3*cosx
y=tgx^3cosx
y=tgx3cosx

The derivative of the function/ y=tgx^3*cosx

Derivative of y=tgx^3*cosx

The solution

You have entered [src]

   3          
tan (x)*cos(x)

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

tan(x)^3*cos(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \tan^{3}{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \tan{\left(x \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
  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:
  
  $\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
$g{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of cosine is negative sine:
  
  $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
The result is: $\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos{\left(x \right)}} - \sin{\left(x \right)} \tan^{3}{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     3                2    /         2   \       
- tan (x)*sin(x) + tan (x)*\3 + 3*tan (x)/*cos(x)

$$\left(3 \tan^{2}{\left(x \right)} + 3\right) \cos{\left(x \right)} \tan^{2}{\left(x \right)} - \sin{\left(x \right)} \tan^{3}{\left(x \right)}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                                                  /             2                                      \                                                        
   3                  2    /       2   \            /       2   \ |/       2   \         4           2    /       2   \|             /       2   \ /         2   \              
tan (x)*sin(x) - 9*tan (x)*\1 + tan (x)/*cos(x) + 6*\1 + tan (x)/*\\1 + tan (x)/  + 2*tan (x) + 7*tan (x)*\1 + tan (x)//*cos(x) - 18*\1 + tan (x)/*\1 + 2*tan (x)/*sin(x)*tan(x)

$$- 18 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)} + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 7 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 2 \tan^{4}{\left(x \right)}\right) \cos{\left(x \right)} - 9 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan^{2}{\left(x \right)} + \sin{\left(x \right)} \tan^{3}{\left(x \right)}$$

Simplify

The graph

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Identical expressions

Derivative of y=tgx^3*cosx

The graph:

Piecewise:

The solution