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Similar expressions
(1+tan(3*x)^(2))*e^(x/2)
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The derivative of the function/ (1+tan(3*x)^(2))*e^(-x/2)

Derivative of (1+tan(3x)^(2))e^(-x/2)

The solution

You have entered [src]

                 -x 
                 ---
/       2     \   2 
\1 + tan (3*x)/*E

$$e^{\frac{\left(-1\right) x}{2}} \left(\tan^{2}{\left(3 x \right)} + 1\right)$$

(1 + tan(3*x)^2)*E^((-x)/2)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \tan^{2}{\left(3 x \right)} + 1$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $\tan^{2}{\left(3 x \right)} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Let $u = \tan{\left(3 x \right)}$ .
  3. Apply the power rule: $u^{2}$ goes to $2 u$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(3 x \right)}$ :
    1. Rewrite the function to be differentiated:
      
      $\tan{\left(3 x \right)} = \frac{\sin{\left(3 x \right)}}{\cos{\left(3 x \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(3 x \right)}$ and $g{\left(x \right)} = \cos{\left(3 x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Let $u = 3 x$ .
      2. The derivative of sine is cosine:
        
        $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $3$
        
        The result of the chain rule is:
        
        $3 \cos{\left(3 x \right)}$
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      1. Let $u = 3 x$ .
      2. The derivative of cosine is negative sine:
        
        $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $3$
        
        The result of the chain rule is:
        
        $- 3 \sin{\left(3 x \right)}$
      Now plug in to the quotient rule:
      
      $\frac{3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}$
    The result of the chain rule is:
    
    $\frac{2 \left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right) \tan{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}$
  The result is: $\frac{2 \left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right) \tan{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}$
$g{\left(x \right)} = e^{\frac{\left(-1\right) x}{2}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \frac{\left(-1\right) x}{2}$ .
2. The derivative of $e^{u}$ is itself.
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\left(-1\right) x}{2}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $-1$
    So, the result is: $- \frac{1}{2}$
  The result of the chain rule is:
  
  $- \frac{e^{\frac{\left(-1\right) x}{2}}}{2}$
The result is: $\frac{2 \left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right) e^{\frac{\left(-1\right) x}{2}} \tan{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} - \frac{\left(\tan^{2}{\left(3 x \right)} + 1\right) e^{\frac{\left(-1\right) x}{2}}}{2}$
Now simplify:

$\frac{\left(12 \tan{\left(3 x \right)} - 1\right) e^{- \frac{x}{2}}}{2 \cos^{2}{\left(3 x \right)}}$

The answer is:

$\frac{\left(12 \tan{\left(3 x \right)} - 1\right) e^{- \frac{x}{2}}}{2 \cos^{2}{\left(3 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                   -x                                   
                   ---                      -x          
  /       2     \   2                       ---         
  \1 + tan (3*x)/*e      /         2     \   2          
- -------------------- + \6 + 6*tan (3*x)/*e   *tan(3*x)
           2

$$- \frac{\left(\tan^{2}{\left(3 x \right)} + 1\right) e^{\frac{\left(-1\right) x}{2}}}{2} + \left(6 \tan^{2}{\left(3 x \right)} + 6\right) e^{\frac{\left(-1\right) x}{2}} \tan{\left(3 x \right)}$$

Simplify

The second derivative [src]

                                                  -x 
                                                  ---
/       2     \ /73                      2     \   2 
\1 + tan (3*x)/*|-- - 6*tan(3*x) + 54*tan (3*x)|*e   
                \4                             /

$$\left(\tan^{2}{\left(3 x \right)} + 1\right) \left(54 \tan^{2}{\left(3 x \right)} - 6 \tan{\left(3 x \right)} + \frac{73}{4}\right) e^{- \frac{x}{2}}$$

Simplify

The third derivative [src]

                                                                                      -x 
                                                                                      ---
/       2     \ /  217         2        9*tan(3*x)       /         2     \         \   2 
\1 + tan (3*x)/*|- --- - 81*tan (3*x) + ---------- + 216*\2 + 3*tan (3*x)/*tan(3*x)|*e   
                \   8                       2                                      /

$$\left(\tan^{2}{\left(3 x \right)} + 1\right) \left(216 \left(3 \tan^{2}{\left(3 x \right)} + 2\right) \tan{\left(3 x \right)} - 81 \tan^{2}{\left(3 x \right)} + \frac{9 \tan{\left(3 x \right)}}{2} - \frac{217}{8}\right) e^{- \frac{x}{2}}$$

Simplify

The graph

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Derivative of (1+tan(3x)^(2))e^(-x/2)

The graph:

Piecewise:

The solution

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Derivative of (1+tan(3*x)^(2))*e^(-x/2)

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Piecewise:

The solution

Derivative of (1+tan(3x)^(2))e^(-x/2)