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The derivative of the function/ tgu/sinv

Derivative of tgu/sinv

The solution

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tan(u)
------
sin(v)

$$\frac{\tan{\left(u \right)}}{\sin{\left(v \right)}}$$

tan(u)/sin(v)

Detail solution

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

$\frac{1}{\sin{\left(v \right)} \cos^{2}{\left(u \right)}}$

The answer is:

$\frac{1}{\sin{\left(v \right)} \cos^{2}{\left(u \right)}}$

The first derivative [src]

       2   
1 + tan (u)
-----------
   sin(v)

$$\frac{\tan^{2}{\left(u \right)} + 1}{\sin{\left(v \right)}}$$

Simplify

The second derivative [src]

  /       2   \       
2*\1 + tan (u)/*tan(u)
----------------------
        sin(v)

$$\frac{2 \left(\tan^{2}{\left(u \right)} + 1\right) \tan{\left(u \right)}}{\sin{\left(v \right)}}$$

Simplify

The third derivative [src]

  /       2   \ /         2   \
2*\1 + tan (u)/*\1 + 3*tan (u)/
-------------------------------
             sin(v)

$$\frac{2 \left(\tan^{2}{\left(u \right)} + 1\right) \left(3 \tan^{2}{\left(u \right)} + 1\right)}{\sin{\left(v \right)}}$$

Simplify