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y=sin(x)*cos^3(x)
The derivative of the function/ y=sin(x)*cos^3(x)

Derivative of y=sin(x)*cos^3(x)

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
          3   
sin(x)*cos (x)
sin⁡(x)cos⁡3(x)\sin{\left(x \right)} \cos^{3}{\left(x \right)} 
sin(x)*cos(x)^3
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)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\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)}

    g(x)=cos⁡3(x)g{\left(x \right)} = \cos^{3}{\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: u3u^{3} goes to 3u23 u^{2}

    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:

      −3sin⁡(x)cos⁡2(x)- 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}

    The result is: −3sin⁡2(x)cos⁡2(x)+cos⁡4(x)- 3 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} + \cos^{4}{\left(x \right)}

  2. Now simplify:

    cos⁡(2x)2+cos⁡(4x)2\frac{\cos{\left(2 x \right)}}{2} + \frac{\cos{\left(4 x \right)}}{2}

The answer is:

cos⁡(2x)2+cos⁡(4x)2\frac{\cos{\left(2 x \right)}}{2} + \frac{\cos{\left(4 x \right)}}{2}

The graph
02468-8-6-4-2-10102-2
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   4           2       2   
cos (x) - 3*cos (x)*sin (x)
−3sin⁡2(x)cos⁡2(x)+cos⁡4(x)- 3 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} + \cos^{4}{\left(x \right)}
Simplify
The second derivative [src] 
/        2           2   \              
\- 10*cos (x) + 6*sin (x)/*cos(x)*sin(x)
(6sin⁡2(x)−10cos⁡2(x))sin⁡(x)cos⁡(x)\left(6 \sin^{2}{\left(x \right)} - 10 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}
Simplify
The third derivative [src] 
     4           2    /       2           2   \        2       2           2    /     2           2   \
- cos (x) - 3*sin (x)*\- 7*cos (x) + 2*sin (x)/ + 9*cos (x)*sin (x) + 9*cos (x)*\- cos (x) + 2*sin (x)/
−3(2sin⁡2(x)−7cos⁡2(x))sin⁡2(x)+9(2sin⁡2(x)−cos⁡2(x))cos⁡2(x)+9sin⁡2(x)cos⁡2(x)−cos⁡4(x)- 3 \left(2 \sin^{2}{\left(x \right)} - 7 \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} + 9 \left(2 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)} + 9 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} - \cos^{4}{\left(x \right)}
Simplify
The graph
Derivative of y=sin(x)*cos^3(x)
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