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cos(sqrt(x))
The derivative of the function/ cos(sqrt(x))

Derivative of cos(sqrt(x))

Function f() - derivative -N order at the point
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The graph:

from to

Piecewise:

The solution

You have entered [src] 
   /  ___\
cos\\/ x /
cos⁡(x)\cos{\left(\sqrt{x} \right)} 
d /   /  ___\\
--\cos\\/ x //
dx
ddxcos⁡(x)\frac{d}{d x} \cos{\left(\sqrt{x} \right)}
Transform
Detail solution

  1. Let u=xu = \sqrt{x}.

  2. The derivative of cosine is negative sine:

    dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

  3. Then, apply the chain rule. Multiply by ddxx\frac{d}{d x} \sqrt{x}:

    1. Apply the power rule: x\sqrt{x} goes to 12x\frac{1}{2 \sqrt{x}}

    The result of the chain rule is:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-2020
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
    /  ___\ 
-sin\\/ x / 
------------
      ___   
  2*\/ x
−sin⁡(x)2x- \frac{\sin{\left(\sqrt{x} \right)}}{2 \sqrt{x}}
Simplify
The second derivative [src] 
   /  ___\      /  ___\
sin\\/ x /   cos\\/ x /
---------- - ----------
    3/2          x     
   x                   
-----------------------
           4
−cos⁡(x)x+sin⁡(x)x324\frac{- \frac{\cos{\left(\sqrt{x} \right)}}{x} + \frac{\sin{\left(\sqrt{x} \right)}}{x^{\frac{3}{2}}}}{4}
Simplify
The third derivative [src] 
   /  ___\        /  ___\        /  ___\
sin\\/ x /   3*sin\\/ x /   3*cos\\/ x /
---------- - ------------ + ------------
    3/2           5/2             2     
   x             x               x      
----------------------------------------
                   8
3cos⁡(x)x2+sin⁡(x)x32−3sin⁡(x)x528\frac{\frac{3 \cos{\left(\sqrt{x} \right)}}{x^{2}} + \frac{\sin{\left(\sqrt{x} \right)}}{x^{\frac{3}{2}}} - \frac{3 \sin{\left(\sqrt{x} \right)}}{x^{\frac{5}{2}}}}{8}
Simplify
The graph
Derivative of cos(sqrt(x))
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