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Derivative of:
Derivative of asin(x)
Derivative of e^2*x
Derivative of x^sin(x)
Derivative of sqrt(x^2+1)
Identical expressions
three ^(two *x)/sin(two *x)
3 to the power of (2 multiply by x) divide by sinus of (2 multiply by x)
three to the power of (two multiply by x) divide by sinus of (two multiply by x)
3(2*x)/sin(2*x)
32*x/sin2*x
3^(2x)/sin(2x)
3(2x)/sin(2x)
32x/sin2x
3^2x/sin2x
3^(2*x) divide by sin(2*x)
What you mean?
3^(2*x)/sin(2*x)
3^(2*x)/((sin(2)*x))
3^(2*x)*2*x/sin(x)
3*2*x/sin(2*x)
3*2*x/(sin(2)*x)
3*2*x*2*x/sin(x)
3^(2*x)*2*x/sin(x)

The derivative of the function/ 3^(2*x)/sin(2*x)

You entered:

3^(2*x)/sin(2*x)

What you mean?

3^(2*x)/sin(2*x)

3^(2*x)/((sin(2)*x))

3^(2*x)*2*x/sin(x)

3*2*x/sin(2*x)

3*2*x/(sin(2)*x)

3*2*x*2*x/sin(x)

3^(2*x)*2*x/sin(x)

Derivative of 3^(2x)/sin(2x)

The solution

You have entered [src]

   2*x  
  3     
--------
sin(2*x)

$$\frac{3^{2 x}}{\sin{\left(2 x \right)}}$$

  /   2*x  \
d |  3     |
--|--------|
dx\sin(2*x)/

$$\frac{d}{d x} \frac{3^{2 x}}{\sin{\left(2 x \right)}}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 2 x$ .
2. $\frac{d}{d u} 3^{u} = 3^{u} \log{\left(3 \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
  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: $2$
  The result of the chain rule is:
  
  $2 \cdot 3^{2 x} \log{\left(3 \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 2 x$ .
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 $\frac{d}{d x} 2 x$ :
  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: $2$
  The result of the chain rule is:
  
  $2 \cos{\left(2 x \right)}$
Now plug in to the quotient rule:

$\frac{2 \cdot 3^{2 x} \log{\left(3 \right)} \sin{\left(2 x \right)} - 2 \cdot 3^{2 x} \cos{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}}$
Now simplify:

$\frac{2 \cdot 9^{x} \left(\log{\left(3 \right)} \sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right)}{\sin^{2}{\left(2 x \right)}}$

The answer is:

$\frac{2 \cdot 9^{x} \left(\log{\left(3 \right)} \sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right)}{\sin^{2}{\left(2 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     2*x               2*x       
  2*3   *cos(2*x)   2*3   *log(3)
- --------------- + -------------
        2              sin(2*x)  
     sin (2*x)

$$\frac{2 \cdot 3^{2 x} \log{\left(3 \right)}}{\sin{\left(2 x \right)}} - \frac{2 \cdot 3^{2 x} \cos{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}}$$

Simplify

The second derivative [src]

       /                   2                         \
   2*x |       2      2*cos (2*x)   2*cos(2*x)*log(3)|
4*3   *|1 + log (3) + ----------- - -----------------|
       |                  2              sin(2*x)    |
       \               sin (2*x)                     /
------------------------------------------------------
                       sin(2*x)

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

Simplify

The third derivative [src]

       /                                       /         2     \                              \
       |                                       |    6*cos (2*x)|                              |
       |                                       |5 + -----------|*cos(2*x)                     |
       |            /         2     \          |        2      |                 2            |
   2*x |   3        |    2*cos (2*x)|          \     sin (2*x) /            3*log (3)*cos(2*x)|
8*3   *|log (3) + 3*|1 + -----------|*log(3) - -------------------------- - ------------------|
       |            |        2      |                   sin(2*x)                 sin(2*x)     |
       \            \     sin (2*x) /                                                         /
-----------------------------------------------------------------------------------------------
                                            sin(2*x)

$$\frac{8 \cdot 3^{2 x} \left(3 \cdot \left(1 + \frac{2 \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}}\right) \log{\left(3 \right)} - \frac{\left(5 + \frac{6 \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}}\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}} + \log{\left(3 \right)}^{3} - \frac{3 \log{\left(3 \right)}^{2} \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}}\right)}{\sin{\left(2 x \right)}}$$

Simplify

The graph

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What you mean?

Derivative of 3^(2x)/sin(2x)

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

What you mean?

You entered:

What you mean?

Derivative of 3^(2*x)/sin(2*x)

The graph:

Piecewise:

The solution

Derivative of 3^(2x)/sin(2x)