Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of 6
Derivative of 2/sqrt(x)
Derivative of x/(x+1)
Derivative of x*e^(1/x)
Integral of d{x}:
tan^2(x)
Identical expressions
tan^ two (x)
tangent of squared (x)
tangent of to the power of two (x)
tan2(x)
tan2x
tan²(x)
tan to the power of 2(x)
tan^2x
Similar expressions
atan^2x
cotan^2x-cos^2x-cotan^2x

The derivative of the function/ tan^2(x)

Derivative of tan^2(x)

The solution

You have entered [src]

   2   
tan (x)

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

d /   2   \
--\tan (x)/
dx

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

Transform

Detail solution

Let $u = \tan{\left(x \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
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 of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /       2   \ /         2   \       
8*\1 + tan (x)/*\2 + 3*tan (x)/*tan(x)

$$8 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}$$

Simplify

The graph

Derivative of tan^2(x)