Derivative of cosxsinxtgx

The solution

You have entered [src]

cos(x)*sin(x)*tan(x)

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

(cos(x)*sin(x))*tan(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sin{\left(x \right)} \cos{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result is: $- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}$
$g{\left(x \right)} = \tan{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result is: $\left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)} + \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/   2         2   \          /       2   \              
\cos (x) - sin (x)/*tan(x) + \1 + tan (x)/*cos(x)*sin(x)

$$\left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)}$$

Simplify

The second derivative [src]

  /  /       2   \ /   2         2   \                            /       2   \                     \
2*\- \1 + tan (x)/*\sin (x) - cos (x)/ - 2*cos(x)*sin(x)*tan(x) + \1 + tan (x)/*cos(x)*sin(x)*tan(x)/

$$2 \left(- \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)} - 2 \sin{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)}\right)$$

Simplify

The third derivative [src]

  /  /   2         2   \            /       2   \                   /       2   \ /   2         2   \          /       2   \ /         2   \              \
2*\2*\sin (x) - cos (x)/*tan(x) - 6*\1 + tan (x)/*cos(x)*sin(x) - 3*\1 + tan (x)/*\sin (x) - cos (x)/*tan(x) + \1 + tan (x)/*\1 + 3*tan (x)/*cos(x)*sin(x)/

$$2 \left(- 3 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} - 6 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)}\right)$$

Simplify

Other calculators

How to use it?

Identical expressions

Derivative of cosxsinxtgx

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Derivative of cosx*sinx*tgx

The graph:

Piecewise:

The solution

Derivative of cosxsinxtgx