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

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

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

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

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

    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:

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

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

  2. Now simplify:

    (3−4sin⁡2(x))sin⁡2(x)\left(3 - 4 \sin^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)}

The answer is:

(3−4sin⁡2(x))sin⁡2(x)\left(3 - 4 \sin^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)}

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   
- sin (x) + 3*cos (x)*sin (x)
−sin⁡4(x)+3sin⁡2(x)cos⁡2(x)- \sin^{4}{\left(x \right)} + 3 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}
Simplify
The second derivative [src] 
 /       2            2   \              
-\- 6*cos (x) + 10*sin (x)/*cos(x)*sin(x)
−(10sin⁡2(x)−6cos⁡2(x))sin⁡(x)cos⁡(x)- \left(10 \sin^{2}{\left(x \right)} - 6 \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   \
sin (x) - 9*cos (x)*sin (x) - 3*cos (x)*\- 2*cos (x) + 7*sin (x)/ + 9*sin (x)*\sin (x) - 2*cos (x)/
9(sin⁡2(x)−2cos⁡2(x))sin⁡2(x)−3(7sin⁡2(x)−2cos⁡2(x))cos⁡2(x)+sin⁡4(x)−9sin⁡2(x)cos⁡2(x)9 \left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} - 3 \left(7 \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)} + \sin^{4}{\left(x \right)} - 9 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}
Simplify
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