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The derivative of the function/ logcosxsinx

Derivative of logcosxsinx

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

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The solution

You have entered [src] 
log(cos(x))*x*sin(x)
xlog⁡(cos⁡(x))sin⁡(x)x \log{\left(\cos{\left(x \right)} \right)} \sin{\left(x \right)} 
(log(cos(x))*x)*sin(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)=xlog⁡(cos⁡(x))f{\left(x \right)} = x \log{\left(\cos{\left(x \right)} \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    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)=log⁡(cos⁡(x))f{\left(x \right)} = \log{\left(\cos{\left(x \right)} \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Let u=cos⁡(x)u = \cos{\left(x \right)}.

      2. The derivative of log⁡(u)\log{\left(u \right)} is 1u\frac{1}{u}.

      3. Then, apply the chain rule. Multiply by ddxcos⁡(x)\frac{d}{d x} \cos{\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 of the chain rule is:

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

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

      1. Apply the power rule: xx goes to 11

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

    g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\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 is: xlog⁡(cos⁡(x))cos⁡(x)+(−xsin⁡(x)cos⁡(x)+log⁡(cos⁡(x)))sin⁡(x)x \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} + \left(- \frac{x \sin{\left(x \right)}}{\cos{\left(x \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right) \sin{\left(x \right)}

  2. Now simplify:

    xlog⁡(cos⁡(x))cos⁡2(x)+xcos⁡(2x)2−x2+log⁡(cos⁡(x))sin⁡(2x)2cos⁡(x)\frac{x \log{\left(\cos{\left(x \right)} \right)} \cos^{2}{\left(x \right)} + \frac{x \cos{\left(2 x \right)}}{2} - \frac{x}{2} + \frac{\log{\left(\cos{\left(x \right)} \right)} \sin{\left(2 x \right)}}{2}}{\cos{\left(x \right)}}

The answer is:

xlog⁡(cos⁡(x))cos⁡2(x)+xcos⁡(2x)2−x2+log⁡(cos⁡(x))sin⁡(2x)2cos⁡(x)\frac{x \log{\left(\cos{\left(x \right)} \right)} \cos^{2}{\left(x \right)} + \frac{x \cos{\left(2 x \right)}}{2} - \frac{x}{2} + \frac{\log{\left(\cos{\left(x \right)} \right)} \sin{\left(2 x \right)}}{2}}{\cos{\left(x \right)}}

The graph
02468-8-6-4-2-1010-250250
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
/  x*sin(x)              \                              
|- -------- + log(cos(x))|*sin(x) + x*cos(x)*log(cos(x))
\   cos(x)               /
xlog⁡(cos⁡(x))cos⁡(x)+(−xsin⁡(x)cos⁡(x)+log⁡(cos⁡(x)))sin⁡(x)x \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} + \left(- \frac{x \sin{\left(x \right)}}{\cos{\left(x \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right) \sin{\left(x \right)}
Simplify
The second derivative [src] 
 //  /       2   \           \                                                                   \
 ||  |    sin (x)|   2*sin(x)|            /               x*sin(x)\                              |
-||x*|1 + -------| + --------|*sin(x) + 2*|-log(cos(x)) + --------|*cos(x) + x*log(cos(x))*sin(x)|
 ||  |       2   |    cos(x) |            \                cos(x) /                              |
 \\  \    cos (x)/           /                                                                   /
−(xlog⁡(cos⁡(x))sin⁡(x)+(x(sin⁡2(x)cos⁡2(x)+1)+2sin⁡(x)cos⁡(x))sin⁡(x)+2(xsin⁡(x)cos⁡(x)−log⁡(cos⁡(x)))cos⁡(x))- (x \log{\left(\cos{\left(x \right)} \right)} \sin{\left(x \right)} + \left(x \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}\right) \sin{\left(x \right)} + 2 \left(\frac{x \sin{\left(x \right)}}{\cos{\left(x \right)}} - \log{\left(\cos{\left(x \right)} \right)}\right) \cos{\left(x \right)})
Simplify
The third derivative [src] 
    /  /       2   \           \                                                                      /       2   \                        
    |  |    sin (x)|   2*sin(x)|            /               x*sin(x)\                                 |    sin (x)| /    2*x*sin(x)\       
- 3*|x*|1 + -------| + --------|*cos(x) + 3*|-log(cos(x)) + --------|*sin(x) - x*cos(x)*log(cos(x)) - |1 + -------|*|3 + ----------|*sin(x)
    |  |       2   |    cos(x) |            \                cos(x) /                                 |       2   | \      cos(x)  /       
    \  \    cos (x)/           /                                                                      \    cos (x)/
−xlog⁡(cos⁡(x))cos⁡(x)−3(x(sin⁡2(x)cos⁡2(x)+1)+2sin⁡(x)cos⁡(x))cos⁡(x)−(sin⁡2(x)cos⁡2(x)+1)(2xsin⁡(x)cos⁡(x)+3)sin⁡(x)+3(xsin⁡(x)cos⁡(x)−log⁡(cos⁡(x)))sin⁡(x)- x \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} - 3 \left(x \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}\right) \cos{\left(x \right)} - \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \left(\frac{2 x \sin{\left(x \right)}}{\cos{\left(x \right)}} + 3\right) \sin{\left(x \right)} + 3 \left(\frac{x \sin{\left(x \right)}}{\cos{\left(x \right)}} - \log{\left(\cos{\left(x \right)} \right)}\right) \sin{\left(x \right)}
Simplify
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