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The derivative of the function/ sqrt(x)*(cosx)

Derivative of sqrt(x)*(cosx)

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

from to

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The solution

You have entered [src] 
  ___       
\/ x *cos(x)
xcos⁡(x)\sqrt{x} \cos{\left(x \right)} 
sqrt(x)*cos(x)
Detail solution

  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

    f(x)=xf{\left(x \right)} = \sqrt{x}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

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

    g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. The derivative of cosine is negative sine:

      ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

    The result is: −xsin⁡(x)+cos⁡(x)2x- \sqrt{x} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{2 \sqrt{x}}

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-10105-5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
 cos(x)     ___       
------- - \/ x *sin(x)
    ___               
2*\/ x
−xsin⁡(x)+cos⁡(x)2x- \sqrt{x} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{2 \sqrt{x}}
Simplify
The second derivative [src] 
 /  ___          sin(x)   cos(x)\
-|\/ x *cos(x) + ------ + ------|
 |                 ___       3/2|
 \               \/ x     4*x   /
−(xcos⁡(x)+sin⁡(x)x+cos⁡(x)4x32)- (\sqrt{x} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{\sqrt{x}} + \frac{\cos{\left(x \right)}}{4 x^{\frac{3}{2}}})
Simplify
The third derivative [src] 
  ___          3*cos(x)   3*sin(x)   3*cos(x)
\/ x *sin(x) - -------- + -------- + --------
                   ___        3/2        5/2 
               2*\/ x      4*x        8*x
xsin⁡(x)−3cos⁡(x)2x+3sin⁡(x)4x32+3cos⁡(x)8x52\sqrt{x} \sin{\left(x \right)} - \frac{3 \cos{\left(x \right)}}{2 \sqrt{x}} + \frac{3 \sin{\left(x \right)}}{4 x^{\frac{3}{2}}} + \frac{3 \cos{\left(x \right)}}{8 x^{\frac{5}{2}}}
Simplify
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