Mister Exam

Other calculators

The derivative of the function/ 2x-3/sinx/4

Derivative of 2x-3/sinx/4

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
      /  3   \
      |------|
      \sin(x)/
2*x - --------
         4
2x−31sin⁡(x)42 x - \frac{3 \frac{1}{\sin{\left(x \right)}}}{4} 
2*x - 3/sin(x)/4
Detail solution

  1. Differentiate 2x−31sin⁡(x)42 x - \frac{3 \frac{1}{\sin{\left(x \right)}}}{4} term by term:

    1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Apply the power rule: xx goes to 11

      So, the result is: 22

    2. 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. Let u=sin⁡(x)u = \sin{\left(x \right)}.

        2. Apply the power rule: 1u\frac{1}{u} goes to −1u2- \frac{1}{u^{2}}

        3. Then, apply the chain rule. Multiply by ddxsin⁡(x)\frac{d}{d x} \sin{\left(x \right)}:

          1. The derivative of sine is cosine:

            ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

          The result of the chain rule is:

          −cos⁡(x)sin⁡2(x)- \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}

        So, the result is: −3cos⁡(x)sin⁡2(x)- \frac{3 \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}

      So, the result is: 3cos⁡(x)4sin⁡2(x)\frac{3 \cos{\left(x \right)}}{4 \sin^{2}{\left(x \right)}}

    The result is: 2+3cos⁡(x)4sin⁡2(x)2 + \frac{3 \cos{\left(x \right)}}{4 \sin^{2}{\left(x \right)}}

The answer is:

2+3cos⁡(x)4sin⁡2(x)2 + \frac{3 \cos{\left(x \right)}}{4 \sin^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-1000010000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
     3*cos(x)
2 + ---------
         2   
    4*sin (x)
2+3cos⁡(x)4sin⁡2(x)2 + \frac{3 \cos{\left(x \right)}}{4 \sin^{2}{\left(x \right)}}
Simplify
The second derivative [src] 
   /         2   \
   |    2*cos (x)|
-3*|1 + ---------|
   |        2    |
   \     sin (x) /
------------------
     4*sin(x)
−3(1+2cos⁡2(x)sin⁡2(x))4sin⁡(x)- \frac{3 \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right)}{4 \sin{\left(x \right)}}
Simplify
The third derivative [src] 
  /         2   \       
  |    6*cos (x)|       
3*|5 + ---------|*cos(x)
  |        2    |       
  \     sin (x) /       
------------------------
            2           
       4*sin (x)
3(5+6cos⁡2(x)sin⁡2(x))cos⁡(x)4sin⁡2(x)\frac{3 \left(5 + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{4 \sin^{2}{\left(x \right)}}
Simplify
    © Mister Exam – Calculator