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y=x^sinx+cosx
The derivative of the function/ y=x^sinx+cosx

Derivative of y=x^sinx+cosx

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

The graph:

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The solution

You have entered [src] 
 sin(x)         
x       + cos(x)
xsin⁡(x)+cos⁡(x)x^{\sin{\left(x \right)}} + \cos{\left(x \right)} 
x^sin(x) + cos(x)
Detail solution

  1. Differentiate xsin⁡(x)+cos⁡(x)x^{\sin{\left(x \right)}} + \cos{\left(x \right)} term by term:

    1. Don't know the steps in finding this derivative.

      But the derivative is

      (log⁡(sin⁡(x))+1)sin⁡sin⁡(x)(x)\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \sin^{\sin{\left(x \right)}}{\left(x \right)}

    2. 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: (log⁡(sin⁡(x))+1)sin⁡sin⁡(x)(x)−sin⁡(x)\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \sin^{\sin{\left(x \right)}}{\left(x \right)} - \sin{\left(x \right)}

The answer is:

(log⁡(sin⁡(x))+1)sin⁡sin⁡(x)(x)−sin⁡(x)\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \sin^{\sin{\left(x \right)}}{\left(x \right)} - \sin{\left(x \right)}

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) /sin(x)                \
-sin(x) + x      *|------ + cos(x)*log(x)|
                  \  x                   /
xsin⁡(x)(log⁡(x)cos⁡(x)+sin⁡(x)x)−sin⁡(x)x^{\sin{\left(x \right)}} \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) - \sin{\left(x \right)}
Simplify
The second derivative [src] 
                                          2                                              
           sin(x) /sin(x)                \     sin(x) /sin(x)                   2*cos(x)\
-cos(x) + x      *|------ + cos(x)*log(x)|  - x      *|------ + log(x)*sin(x) - --------|
                  \  x                   /            |   2                        x    |
                                                      \  x                              /
xsin⁡(x)(log⁡(x)cos⁡(x)+sin⁡(x)x)2−xsin⁡(x)(log⁡(x)sin⁡(x)−2cos⁡(x)x+sin⁡(x)x2)−cos⁡(x)x^{\sin{\left(x \right)}} \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{2} - x^{\sin{\left(x \right)}} \left(\log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x} + \frac{\sin{\left(x \right)}}{x^{2}}\right) - \cos{\left(x \right)}
Simplify
The third derivative [src] 
                                3                                                                                                                                             
 sin(x) /sin(x)                \     sin(x) /                2*sin(x)   3*sin(x)   3*cos(x)\      sin(x) /sin(x)                \ /sin(x)                   2*cos(x)\         
x      *|------ + cos(x)*log(x)|  - x      *|cos(x)*log(x) - -------- + -------- + --------| - 3*x      *|------ + cos(x)*log(x)|*|------ + log(x)*sin(x) - --------| + sin(x)
        \  x                   /            |                    3         x           2   |             \  x                   / |   2                        x    |         
                                            \                   x                     x    /                                      \  x                              /
xsin⁡(x)(log⁡(x)cos⁡(x)+sin⁡(x)x)3−3xsin⁡(x)(log⁡(x)cos⁡(x)+sin⁡(x)x)(log⁡(x)sin⁡(x)−2cos⁡(x)x+sin⁡(x)x2)−xsin⁡(x)(log⁡(x)cos⁡(x)+3sin⁡(x)x+3cos⁡(x)x2−2sin⁡(x)x3)+sin⁡(x)x^{\sin{\left(x \right)}} \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{3} - 3 x^{\sin{\left(x \right)}} \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \left(\log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x} + \frac{\sin{\left(x \right)}}{x^{2}}\right) - x^{\sin{\left(x \right)}} \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{3 \sin{\left(x \right)}}{x} + \frac{3 \cos{\left(x \right)}}{x^{2}} - \frac{2 \sin{\left(x \right)}}{x^{3}}\right) + \sin{\left(x \right)}
Simplify
The graph
Derivative of y=x^sinx+cosx
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