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The derivative of the function/ y=sin³x*cos²x

Derivative of y=sin³x*cos²x

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

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The solution

You have entered [src] 
   3       2   
sin (x)*cos (x)
sin⁡3(x)cos⁡2(x)\sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)} 
d /   3       2   \
--\sin (x)*cos (x)/
dx
ddxsin⁡3(x)cos⁡2(x)\frac{d}{d x} \sin^{3}{\left(x \right)} \cos^{2}{\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)=sin⁡3(x)f{\left(x \right)} = \sin^{3}{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\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)}

    g(x)=cos⁡2(x)g{\left(x \right)} = \cos^{2}{\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: u2u^{2} goes to 2u2 u

    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:

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

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

  2. Now simplify:

    (3−5sin⁡2(x))sin⁡2(x)cos⁡(x)\left(3 - 5 \sin^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)}

The answer is:

(3−5sin⁡2(x))sin⁡2(x)cos⁡(x)\left(3 - 5 \sin^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)}

The first derivative [src] 
       4                  3       2   
- 2*sin (x)*cos(x) + 3*cos (x)*sin (x)
−2sin⁡4(x)cos⁡(x)+3sin⁡2(x)cos⁡3(x)- 2 \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 3 \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)}
Simplify
The second derivative [src] 
/        2       2           2    /   2           2   \        2    /   2         2   \\       
\- 12*cos (x)*sin (x) - 3*cos (x)*\sin (x) - 2*cos (x)/ + 2*sin (x)*\sin (x) - cos (x)//*sin(x)
(−3(sin⁡2(x)−2cos⁡2(x))cos⁡2(x)+2(sin⁡2(x)−cos⁡2(x))sin⁡2(x)−12sin⁡2(x)cos⁡2(x))sin⁡(x)\left(- 3 \left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)} + 2 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} - 12 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}
Simplify
The third derivative [src] 
/     4           2    /       2           2   \         2    /   2         2   \         2    /   2           2   \\       
\8*sin (x) - 3*cos (x)*\- 2*cos (x) + 7*sin (x)/ + 18*sin (x)*\sin (x) - cos (x)/ + 18*sin (x)*\sin (x) - 2*cos (x)//*cos(x)
(18(sin⁡2(x)−2cos⁡2(x))sin⁡2(x)+18(sin⁡2(x)−cos⁡2(x))sin⁡2(x)−3⋅(7sin⁡2(x)−2cos⁡2(x))cos⁡2(x)+8sin⁡4(x))cos⁡(x)\left(18 \left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} + 18 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} - 3 \cdot \left(7 \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)} + 8 \sin^{4}{\left(x \right)}\right) \cos{\left(x \right)}
Simplify
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