Derivative of y=tgx*sin^2x

The solution

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          2   
tan(x)*sin (x)

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

tan(x)*sin(x)^2

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \tan{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
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:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
$g{\left(x \right)} = \sin^{2}{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $2 \sin{\left(x \right)} \cos{\left(x \right)}$
The result is: $\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2 \sin{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of y=tgx*sin^2x

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The solution