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cos2x*lnx
The derivative of the function/ cos2x*lnx

Derivative of cos2x*lnx

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

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The solution

You have entered [src] 
cos(2*x)*log(x)
log⁡(x)cos⁡(2x)\log{\left(x \right)} \cos{\left(2 x \right)} 
d                  
--(cos(2*x)*log(x))
dx
ddxlog⁡(x)cos⁡(2x)\frac{d}{d x} \log{\left(x \right)} \cos{\left(2 x \right)}
Transform
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)=cos⁡(2x)f{\left(x \right)} = \cos{\left(2 x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Let u=2xu = 2 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 ddx2x\frac{d}{d x} 2 x:

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

        1. Apply the power rule: xx goes to 11

        So, the result is: 22

      The result of the chain rule is:

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

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

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

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

The answer is:

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

The graph
02468-8-6-4-2-1010-2020
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
cos(2*x)                    
-------- - 2*log(x)*sin(2*x)
   x
−2log⁡(x)sin⁡(2x)+cos⁡(2x)x- 2 \log{\left(x \right)} \sin{\left(2 x \right)} + \frac{\cos{\left(2 x \right)}}{x}
Simplify
The second derivative [src] 
 /cos(2*x)   4*sin(2*x)                    \
-|-------- + ---------- + 4*cos(2*x)*log(x)|
 |    2          x                         |
 \   x                                     /
−(4log⁡(x)cos⁡(2x)+4sin⁡(2x)x+cos⁡(2x)x2)- (4 \log{\left(x \right)} \cos{\left(2 x \right)} + \frac{4 \sin{\left(2 x \right)}}{x} + \frac{\cos{\left(2 x \right)}}{x^{2}})
Simplify
The third derivative [src] 
  /cos(2*x)   6*cos(2*x)   3*sin(2*x)                    \
2*|-------- - ---------- + ---------- + 4*log(x)*sin(2*x)|
  |    3          x             2                        |
  \   x                        x                         /
2⋅(4log⁡(x)sin⁡(2x)−6cos⁡(2x)x+3sin⁡(2x)x2+cos⁡(2x)x3)2 \cdot \left(4 \log{\left(x \right)} \sin{\left(2 x \right)} - \frac{6 \cos{\left(2 x \right)}}{x} + \frac{3 \sin{\left(2 x \right)}}{x^{2}} + \frac{\cos{\left(2 x \right)}}{x^{3}}\right)
Simplify
The graph
Derivative of cos2x*lnx
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