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y=ln*sin^4x
The derivative of the function/ y=ln*sin^4x

Derivative of y=ln*sin^4x

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

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The solution

You have entered [src] 
          4   
log(x)*sin (x)
log⁡(x)sin⁡4(x)\log{\left(x \right)} \sin^{4}{\left(x \right)} 
d /          4   \
--\log(x)*sin (x)/
dx
ddxlog⁡(x)sin⁡4(x)\frac{d}{d x} \log{\left(x \right)} \sin^{4}{\left(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)=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)=sin⁡4(x)g{\left(x \right)} = \sin^{4}{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

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

    2. Apply the power rule: u4u^{4} goes to 4u34 u^{3}

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

      4sin⁡3(x)cos⁡(x)4 \sin^{3}{\left(x \right)} \cos{\left(x \right)}

    The result is: 4log⁡(x)sin⁡3(x)cos⁡(x)+sin⁡4(x)x4 \log{\left(x \right)} \sin^{3}{\left(x \right)} \cos{\left(x \right)} + \frac{\sin^{4}{\left(x \right)}}{x}

  2. Now simplify:

    (4xlog⁡(x)cos⁡(x)+sin⁡(x))sin⁡3(x)x\frac{\left(4 x \log{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{3}{\left(x \right)}}{x}

The answer is:

(4xlog⁡(x)cos⁡(x)+sin⁡(x))sin⁡3(x)x\frac{\left(4 x \log{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{3}{\left(x \right)}}{x}

The graph
02468-8-6-4-2-10105-5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   4                             
sin (x)        3                 
------- + 4*sin (x)*cos(x)*log(x)
   x
4log⁡(x)sin⁡3(x)cos⁡(x)+sin⁡4(x)x4 \log{\left(x \right)} \sin^{3}{\left(x \right)} \cos{\left(x \right)} + \frac{\sin^{4}{\left(x \right)}}{x}
Simplify
The second derivative [src] 
        /     2                                                      \
   2    |  sin (x)     /   2           2   \          8*cos(x)*sin(x)|
sin (x)*|- ------- - 4*\sin (x) - 3*cos (x)/*log(x) + ---------------|
        |      2                                             x       |
        \     x                                                      /
(−4(sin⁡2(x)−3cos⁡2(x))log⁡(x)+8sin⁡(x)cos⁡(x)x−sin⁡2(x)x2)sin⁡2(x)\left(- 4 \left(\sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right) \log{\left(x \right)} + \frac{8 \sin{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right) \sin^{2}{\left(x \right)}
Simplify
The third derivative [src] 
  /   3        /   2           2   \               2                                                      \       
  |sin (x)   6*\sin (x) - 3*cos (x)/*sin(x)   6*sin (x)*cos(x)     /       2           2   \              |       
2*|------- - ------------------------------ - ---------------- - 4*\- 3*cos (x) + 5*sin (x)/*cos(x)*log(x)|*sin(x)
  |    3                   x                          2                                                   |       
  \   x                                              x                                                    /
2(−4⋅(5sin⁡2(x)−3cos⁡2(x))log⁡(x)cos⁡(x)−6(sin⁡2(x)−3cos⁡2(x))sin⁡(x)x−6sin⁡2(x)cos⁡(x)x2+sin⁡3(x)x3)sin⁡(x)2 \left(- 4 \cdot \left(5 \sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right) \log{\left(x \right)} \cos{\left(x \right)} - \frac{6 \left(\sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{x} - \frac{6 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{\sin^{3}{\left(x \right)}}{x^{3}}\right) \sin{\left(x \right)}
Simplify
The graph
Derivative of y=ln*sin^4x
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