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y=ln*cos²5x
The derivative of the function/ y=ln*cos²5x

Derivative of y=ln*cos²5x

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

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The solution

You have entered [src] 
          25   
log(x)*cos  (x)
log⁡(x)cos⁡25(x)\log{\left(x \right)} \cos^{25}{\left(x \right)} 
log(x)*cos(x)^25
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)=log⁡(x)f{\left(x \right)} = \log{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

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

    g(x)=cos⁡25(x)g{\left(x \right)} = \cos^{25}{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

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

    2. Apply the power rule: u25u^{25} goes to 25u2425 u^{24}

    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:

      −25sin⁡(x)cos⁡24(x)- 25 \sin{\left(x \right)} \cos^{24}{\left(x \right)}

    The result is: −25log⁡(x)sin⁡(x)cos⁡24(x)+cos⁡25(x)x- 25 \log{\left(x \right)} \sin{\left(x \right)} \cos^{24}{\left(x \right)} + \frac{\cos^{25}{\left(x \right)}}{x}

  2. Now simplify:

    (−25xlog⁡(x)sin⁡(x)+cos⁡(x))cos⁡24(x)x\frac{\left(- 25 x \log{\left(x \right)} \sin{\left(x \right)} + \cos{\left(x \right)}\right) \cos^{24}{\left(x \right)}}{x}

The answer is:

(−25xlog⁡(x)sin⁡(x)+cos⁡(x))cos⁡24(x)x\frac{\left(- 25 x \log{\left(x \right)} \sin{\left(x \right)} + \cos{\left(x \right)}\right) \cos^{24}{\left(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] 
   25                               
cos  (x)         24                 
-------- - 25*cos  (x)*log(x)*sin(x)
   x
−25log⁡(x)sin⁡(x)cos⁡24(x)+cos⁡25(x)x- 25 \log{\left(x \right)} \sin{\left(x \right)} \cos^{24}{\left(x \right)} + \frac{\cos^{25}{\left(x \right)}}{x}
Simplify
The second derivative [src] 
         /     2                                                           \
   23    |  cos (x)      /     2            2   \          50*cos(x)*sin(x)|
cos  (x)*|- ------- + 25*\- cos (x) + 24*sin (x)/*log(x) - ----------------|
         |      2                                                 x        |
         \     x                                                           /
(25(24sin⁡2(x)−cos⁡2(x))log⁡(x)−50sin⁡(x)cos⁡(x)x−cos⁡2(x)x2)cos⁡23(x)\left(25 \left(24 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(x \right)} - \frac{50 \sin{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{\cos^{2}{\left(x \right)}}{x^{2}}\right) \cos^{23}{\left(x \right)}
Simplify
The third derivative [src] 
         /     3                                                         /     2            2   \                2          \
   22    |2*cos (x)      /        2             2   \                 75*\- cos (x) + 24*sin (x)/*cos(x)   75*cos (x)*sin(x)|
cos  (x)*|--------- - 25*\- 73*cos (x) + 552*sin (x)/*log(x)*sin(x) + ---------------------------------- + -----------------|
         |     3                                                                      x                             2       |
         \    x                                                                                                    x        /
(−25(552sin⁡2(x)−73cos⁡2(x))log⁡(x)sin⁡(x)+75(24sin⁡2(x)−cos⁡2(x))cos⁡(x)x+75sin⁡(x)cos⁡2(x)x2+2cos⁡3(x)x3)cos⁡22(x)\left(- 25 \left(552 \sin^{2}{\left(x \right)} - 73 \cos^{2}{\left(x \right)}\right) \log{\left(x \right)} \sin{\left(x \right)} + \frac{75 \left(24 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)}}{x} + \frac{75 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{2}} + \frac{2 \cos^{3}{\left(x \right)}}{x^{3}}\right) \cos^{22}{\left(x \right)}
Simplify
The graph
Derivative of y=ln*cos²5x
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