Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of -2/x
Derivative of 1/cos(x)
Derivative of 4/x
Derivative of 1/(1-x)
Identical expressions
five *tan(x)^ three +sin(four *x)*e^((-x)/ seven)
5 multiply by tangent of (x) cubed plus sinus of (4 multiply by x) multiply by e to the power of (( minus x) divide by 7)
five multiply by tangent of (x) to the power of three plus sinus of (four multiply by x) multiply by e to the power of (( minus x) divide by seven)
5*tan(x)3+sin(4*x)*e((-x)/7)
5*tanx3+sin4*x*e-x/7
5*tan(x)³+sin(4*x)*e^((-x)/7)
5*tan(x) to the power of 3+sin(4*x)*e to the power of ((-x)/7)
5tan(x)^3+sin(4x)e^((-x)/7)
5tan(x)3+sin(4x)e((-x)/7)
5tanx3+sin4xe-x/7
5tanx^3+sin4xe^-x/7
5*tan(x)^3+sin(4*x)*e^((-x) divide by 7)
Similar expressions
5*tan(x)^3-sin(4*x)*e^((-x)/7)
5*tan(x)^3+sin(4*x)*e^((x)/7)

The derivative of the function/ 5*tan(x)^3+sin(4*x)*e^((-x)/7)

Derivative of 5tan(x)^3+sin(4x)*e^((-x)/7)

The solution

You have entered [src]

                      -x 
                      ---
     3                 7 
5*tan (x) + sin(4*x)*e

$$e^{\frac{\left(-1\right) x}{7}} \sin{\left(4 x \right)} + 5 \tan^{3}{\left(x \right)}$$

  /                      -x \
  |                      ---|
d |     3                 7 |
--\5*tan (x) + sin(4*x)*e   /
dx

$$\frac{d}{d x} \left(e^{\frac{\left(-1\right) x}{7}} \sin{\left(4 x \right)} + 5 \tan^{3}{\left(x \right)}\right)$$

Transform

Detail solution

Differentiate $e^{\frac{\left(-1\right) x}{7}} \sin{\left(4 x \right)} + 5 \tan^{3}{\left(x \right)}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  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:
    
    $\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  So, the result is: $\frac{15 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
2. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \sin{\left(4 x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 4 x$ .
  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 $\frac{d}{d x} 4 x$ :
    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: $4$
    The result of the chain rule is:
    
    $4 \cos{\left(4 x \right)}$
  $g{\left(x \right)} = e^{\frac{\left(-1\right) x}{7}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = \frac{\left(-1\right) x}{7}$ .
  2. The derivative of $e^{u}$ is itself.
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\left(-1\right) x}{7}$ :
    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.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $-1$
      So, the result is: $- \frac{1}{7}$
    The result of the chain rule is:
    
    $- \frac{e^{\frac{\left(-1\right) x}{7}}}{7}$
  The result is: $- \frac{e^{\frac{\left(-1\right) x}{7}} \sin{\left(4 x \right)}}{7} + 4 e^{\frac{\left(-1\right) x}{7}} \cos{\left(4 x \right)}$
The result is: $\frac{15 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{e^{\frac{\left(-1\right) x}{7}} \sin{\left(4 x \right)}}{7} + 4 e^{\frac{\left(-1\right) x}{7}} \cos{\left(4 x \right)}$
Now simplify:

$\frac{15 \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{e^{- \frac{x}{7}} \sin{\left(4 x \right)}}{7} + 4 e^{- \frac{x}{7}} \cos{\left(4 x \right)}$

The answer is:

$\frac{15 \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{e^{- \frac{x}{7}} \sin{\left(4 x \right)}}{7} + 4 e^{- \frac{x}{7}} \cos{\left(4 x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                                               -x          
            -x                                 ---         
            ---                                 7          
             7         2    /         2   \   e   *sin(4*x)
4*cos(4*x)*e    + 5*tan (x)*\3 + 3*tan (x)/ - -------------
                                                    7

$$5 \cdot \left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)} - \frac{e^{\frac{\left(-1\right) x}{7}} \sin{\left(4 x \right)}}{7} + 4 e^{\frac{\left(-1\right) x}{7}} \cos{\left(4 x \right)}$$

Simplify

The second derivative [src]

                                                           -x                         -x 
                                                           ---                        ---
                2                                           7                          7 
   /       2   \                 3    /       2   \   783*e   *sin(4*x)   8*cos(4*x)*e   
30*\1 + tan (x)/ *tan(x) + 30*tan (x)*\1 + tan (x)/ - ----------------- - ---------------
                                                              49                 7

$$30 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan{\left(x \right)} + 30 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{3}{\left(x \right)} - \frac{783 e^{- \frac{x}{7}} \sin{\left(4 x \right)}}{49} - \frac{8 e^{- \frac{x}{7}} \cos{\left(4 x \right)}}{7}$$

Simplify

The third derivative [src]

                                                                                           -x          -x          
                                                                                           ---         ---         
                3                                               2                           7           7          
   /       2   \          4    /       2   \       /       2   \     2      3124*cos(4*x)*e      2351*e   *sin(4*x)
30*\1 + tan (x)/  + 60*tan (x)*\1 + tan (x)/ + 210*\1 + tan (x)/ *tan (x) - ------------------ + ------------------
                                                                                    49                  343

$$30 \left(\tan^{2}{\left(x \right)} + 1\right)^{3} + 210 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan^{2}{\left(x \right)} + 60 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{4}{\left(x \right)} + \frac{2351 e^{- \frac{x}{7}} \sin{\left(4 x \right)}}{343} - \frac{3124 e^{- \frac{x}{7}} \cos{\left(4 x \right)}}{49}$$

Simplify

The graph

Derivative of 5*tan(x)^3+sin(4*x)*e^((-x)/7)

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of 5tan(x)^3+sin(4x)*e^((-x)/7)

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of 5*tan(x)^3+sin(4*x)*e^((-x)/7)

The graph:

Piecewise:

The solution

Derivative of 5tan(x)^3+sin(4x)*e^((-x)/7)