Derivative of x^2sinxtanx

The solution

You have entered [src]

 2              
x *sin(x)*tan(x)

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

(x^2*sin(x))*tan(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x^{2} \sin{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x^{2}$ goes to $2 x$
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; 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: $x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)}$
$g{\left(x \right)} = \tan{\left(x \right)}$ ; 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: $\frac{x^{2} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \left(x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)}\right) \tan{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

/            2                    \              /       2   \                            2 /       2   \              
\2*sin(x) - x *sin(x) + 4*x*cos(x)/*tan(x) + 2*x*\1 + tan (x)/*(2*sin(x) + x*cos(x)) + 2*x *\1 + tan (x)/*sin(x)*tan(x)

$$2 x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)} + 2 x \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + \left(- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \tan{\left(x \right)}$$

Simplify

The third derivative [src]

  /             2                    \            /       2   \ /            2                    \      2 /       2   \ /         2   \              /       2   \                             
- \-6*cos(x) + x *cos(x) + 6*x*sin(x)/*tan(x) + 3*\1 + tan (x)/*\2*sin(x) - x *sin(x) + 4*x*cos(x)/ + 2*x *\1 + tan (x)/*\1 + 3*tan (x)/*sin(x) + 6*x*\1 + tan (x)/*(2*sin(x) + x*cos(x))*tan(x)

$$2 x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + 6 x \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 3 \left(\tan^{2}{\left(x \right)} + 1\right) \left(- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) - \left(x^{2} \cos{\left(x \right)} + 6 x \sin{\left(x \right)} - 6 \cos{\left(x \right)}\right) \tan{\left(x \right)}$$

Simplify

Other calculators

How to use it?

Identical expressions

Derivative of x^2sinxtanx

The graph:

Piecewise:

The solution