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The derivative of the function/ senx×cosx

Derivative of senx×cosx

The solution

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sin(x)*cos(x)

$$\sin{\left(x \right)} \cos{\left(x \right)}$$

sin(x)*cos(x)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
$g{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of cosine is negative sine:
  
  $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
The result is: $- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}$
Now simplify:

$\cos{\left(2 x \right)}$

The answer is:

$\cos{\left(2 x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2         2   
cos (x) - sin (x)

$$- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}$$

Simplify

The second derivative [src]

-4*cos(x)*sin(x)

$$- 4 \sin{\left(x \right)} \cos{\left(x \right)}$$

Simplify

The third derivative [src]

  /   2         2   \
4*\sin (x) - cos (x)/

$$4 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)$$

Simplify

The graph

Derivative of senx×cosx