Derivative of tan(xsin(x))

The solution

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tan(x*sin(x))

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

tan(x*sin(x))

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(x \sin{\left(x \right)} \right)} = \frac{\sin{\left(x \sin{\left(x \right)} \right)}}{\cos{\left(x \sin{\left(x \right)} \right)}}$
Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sin{\left(x \sin{\left(x \right)} \right)}$ and $g{\left(x \right)} = \cos{\left(x \sin{\left(x \right)} \right)}$ .

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

$\frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin^{2}{\left(x \sin{\left(x \right)} \right)} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos^{2}{\left(x \sin{\left(x \right)} \right)}}{\cos^{2}{\left(x \sin{\left(x \right)} \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of tan(xsin(x))

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The solution