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Derivative of:
Derivative of (x+5)/(x+1)
Derivative of -x/2
Derivative of sin(4x)
Derivative of x*2^x
Identical expressions
(4e^(3x))/(2cos3x)
(4e to the power of (3x)) divide by (2 co sinus of e of 3x)
(4e(3x))/(2cos3x)
4e3x/2cos3x
4e^3x/2cos3x
(4e^(3x)) divide by (2cos3x)

The derivative of the function/ (4e^(3x))/(2cos3x)

Derivative of (4e^(3x))/(2cos3x)

The solution

You have entered [src]

     3*x  
  4*E     
----------
2*cos(3*x)

$$\frac{4 e^{3 x}}{2 \cos{\left(3 x \right)}}$$

(4*E^(3*x))/((2*cos(3*x)))

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = 4 e^{3 x}$ and $g{\left(x \right)} = 2 \cos{\left(3 x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = 3 x$ .
  2. The derivative of $e^{u}$ is itself.
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
    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: $3$
    The result of the chain rule is:
    
    $3 e^{3 x}$
  So, the result is: $12 e^{3 x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  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 $\frac{d}{d x} 3 x$ :
    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: $3$
    The result of the chain rule is:
    
    $- 3 \sin{\left(3 x \right)}$
  So, the result is: $- 6 \sin{\left(3 x \right)}$
Now plug in to the quotient rule:

$\frac{24 e^{3 x} \sin{\left(3 x \right)} + 24 e^{3 x} \cos{\left(3 x \right)}}{4 \cos^{2}{\left(3 x \right)}}$
Now simplify:

$\frac{6 \sqrt{2} e^{3 x} \sin{\left(3 x + \frac{\pi}{4} \right)}}{\cos^{2}{\left(3 x \right)}}$

The answer is:

$\frac{6 \sqrt{2} e^{3 x} \sin{\left(3 x + \frac{\pi}{4} \right)}}{\cos^{2}{\left(3 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                        3*x         
       1       3*x   6*e   *sin(3*x)
12*----------*e    + ---------------
   2*cos(3*x)              2        
                        cos (3*x)

$$\frac{6 e^{3 x} \sin{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + 12 e^{3 x} \frac{1}{2 \cos{\left(3 x \right)}}$$

Simplify

The second derivative [src]

   /                      2     \     
   |    2*sin(3*x)   2*sin (3*x)|  3*x
18*|2 + ---------- + -----------|*e   
   |     cos(3*x)        2      |     
   \                  cos (3*x) /     
--------------------------------------
               cos(3*x)

$$\frac{18 \left(\frac{2 \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + \frac{2 \sin{\left(3 x \right)}}{\cos{\left(3 x \right)}} + 2\right) e^{3 x}}{\cos{\left(3 x \right)}}$$

Simplify

The third derivative [src]

   /                               /         2     \         \     
   |                               |    6*sin (3*x)|         |     
   |                               |5 + -----------|*sin(3*x)|     
   |                      2        |        2      |         |     
   |    3*sin(3*x)   6*sin (3*x)   \     cos (3*x) /         |  3*x
54*|4 + ---------- + ----------- + --------------------------|*e   
   |     cos(3*x)        2                  cos(3*x)         |     
   \                  cos (3*x)                              /     
-------------------------------------------------------------------
                              cos(3*x)

$$\frac{54 \left(\frac{\left(\frac{6 \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + 5\right) \sin{\left(3 x \right)}}{\cos{\left(3 x \right)}} + \frac{6 \sin^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + \frac{3 \sin{\left(3 x \right)}}{\cos{\left(3 x \right)}} + 4\right) e^{3 x}}{\cos{\left(3 x \right)}}$$

Simplify

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Identical expressions

Derivative of (4e^(3x))/(2cos3x)

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Piecewise:

The solution