Mister Exam

Lang:

The derivative of the function/ tanx/2

Derivative of tanx/2

The solution

You have entered [src]

tan(x)
------
  2

$$\frac{\tan{\left(x \right)}}{2}$$

tan(x)/2

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:
  
  $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
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:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
So, the result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)}}$
Now simplify:

$\frac{1}{2 \cos^{2}{\left(x \right)}}$

The answer is:

$\frac{1}{2 \cos^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2   
1   tan (x)
- + -------
2      2

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

/       2   \ /         2   \
\1 + tan (x)/*\1 + 3*tan (x)/

$$\left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right)$$

Simplify

The graph

Derivative of tanx/2