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sin(y)*cos(y)
The derivative of the function/ sin(y)*cos(y)

Derivative of sin(y)*cos(y)

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

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The solution

You have entered [src] 
sin(y)*cos(y)
sin⁡(y)cos⁡(y)\sin{\left(y \right)} \cos{\left(y \right)} 
d                
--(sin(y)*cos(y))
dy
ddysin⁡(y)cos⁡(y)\frac{d}{d y} \sin{\left(y \right)} \cos{\left(y \right)}
Transform
Detail solution

  1. Apply the product rule:

    ddyf(y)g(y)=f(y)ddyg(y)+g(y)ddyf(y)\frac{d}{d y} f{\left(y \right)} g{\left(y \right)} = f{\left(y \right)} \frac{d}{d y} g{\left(y \right)} + g{\left(y \right)} \frac{d}{d y} f{\left(y \right)}

    f(y)=sin⁡(y)f{\left(y \right)} = \sin{\left(y \right)}; to find ddyf(y)\frac{d}{d y} f{\left(y \right)}:

    1. The derivative of sine is cosine:

      ddysin⁡(y)=cos⁡(y)\frac{d}{d y} \sin{\left(y \right)} = \cos{\left(y \right)}

    g(y)=cos⁡(y)g{\left(y \right)} = \cos{\left(y \right)}; to find ddyg(y)\frac{d}{d y} g{\left(y \right)}:

    1. The derivative of cosine is negative sine:

      ddycos⁡(y)=−sin⁡(y)\frac{d}{d y} \cos{\left(y \right)} = - \sin{\left(y \right)}

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

  2. Now simplify:

    cos⁡(2y)\cos{\left(2 y \right)}

The answer is:

cos⁡(2y)\cos{\left(2 y \right)}

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