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y=sqrt*sin^2*x+3cos^3*4x

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  • Identical expressions

  • y=sqrt*sin^ two *x+ three cos^3*4x
  • y equally square root of multiply by sinus of squared multiply by x plus 3 co sinus of e of cubed multiply by 4x
  • y equally square root of multiply by sinus of to the power of two multiply by x plus three co sinus of e of cubed multiply by 4x
  • y=√*sin^2*x+3cos^3*4x
  • y=sqrt*sin2*x+3cos3*4x
  • y=sqrt*sin²*x+3cos³*4x
  • y=sqrt*sin to the power of 2*x+3cos to the power of 3*4x
  • y=sqrtsin^2x+3cos^34x
  • y=sqrtsin2x+3cos34x

  • Similar expressions

  • y=sqrt*sin^2*x-3cos^3*4x
The derivative of the function/ y=sqrt*sin^2*x+3cos^3*4x

Derivative of y=sqrt*sin^2*x+3cos^3*4x

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  ___    2           3     
\/ x *sin (x) + 3*cos (4)*x
$$\sqrt{x} \sin^{2}{\left(x \right)} + x 3 \cos^{3}{\left(4 \right)}$$ 
sqrt(x)*sin(x)^2 + (3*cos(4)^3)*x
Detail solution

  1. Differentiate term by term:

    1. Apply the product rule:

      ; to find :

      1. Apply the power rule: goes to

      ; to find :

      1. Let .

      2. Apply the power rule: goes to

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of sine is cosine:

        The result of the chain rule is:

      The result is:

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

      1. Apply the power rule: goes to

      So, the result is:

    The result is:

  2. Now simplify:

The answer is:

The graph
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
               2                           
     3      sin (x)       ___              
3*cos (4) + ------- + 2*\/ x *cos(x)*sin(x)
                ___                        
            2*\/ x
$$2 \sqrt{x} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \cos^{3}{\left(4 \right)} + \frac{\sin^{2}{\left(x \right)}}{2 \sqrt{x}}$$
Simplify
The second derivative [src] 
                                         2                     
      ___    2          ___    2      sin (x)   2*cos(x)*sin(x)
- 2*\/ x *sin (x) + 2*\/ x *cos (x) - ------- + ---------------
                                          3/2          ___     
                                       4*x           \/ x
$$- 2 \sqrt{x} \sin^{2}{\left(x \right)} + 2 \sqrt{x} \cos^{2}{\left(x \right)} + \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{x}} - \frac{\sin^{2}{\left(x \right)}}{4 x^{\frac{3}{2}}}$$
Simplify
The third derivative [src] 
       2           2           2                                             
  3*sin (x)   3*cos (x)   3*sin (x)       ___                 3*cos(x)*sin(x)
- --------- + --------- + --------- - 8*\/ x *cos(x)*sin(x) - ---------------
      ___         ___          5/2                                    3/2    
    \/ x        \/ x        8*x                                    2*x
$$- 8 \sqrt{x} \sin{\left(x \right)} \cos{\left(x \right)} - \frac{3 \sin^{2}{\left(x \right)}}{\sqrt{x}} + \frac{3 \cos^{2}{\left(x \right)}}{\sqrt{x}} - \frac{3 \sin{\left(x \right)} \cos{\left(x \right)}}{2 x^{\frac{3}{2}}} + \frac{3 \sin^{2}{\left(x \right)}}{8 x^{\frac{5}{2}}}$$
Simplify
The graph
Derivative of y=sqrt*sin^2*x+3cos^3*4x
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