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Derivative of:
Derivative of e^x^x
Derivative of x/(x-4)
Derivative of tan(e^(2*x))
Derivative of sin^3*x
Identical expressions
tan(e^(two *x))
tangent of (e to the power of (2 multiply by x))
tangent of (e to the power of (two multiply by x))
tan(e(2*x))
tane2*x
tan(e^(2x))
tan(e(2x))
tane2x
tane^2x

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

Derivative of tan(e^(2*x))

The solution

You have entered [src]

   / 2*x\
tan\E   /

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

tan(E^(2*x))

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(e^{2 x} \right)} = \frac{\sin{\left(e^{2 x} \right)}}{\cos{\left(e^{2 x} \right)}}$
Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(e^{2 x} \right)}$ and $g{\left(x \right)} = \cos{\left(e^{2 x} \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = e^{2 x}$ .
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 $\frac{d}{d x} e^{2 x}$ :
  1. Let $u = 2 x$ .
  2. The derivative of $e^{u}$ is itself.
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $2$
    The result of the chain rule is:
    
    $2 e^{2 x}$
  The result of the chain rule is:
  
  $2 e^{2 x} \cos{\left(e^{2 x} \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = e^{2 x}$ .
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 $\frac{d}{d x} e^{2 x}$ :
  1. Let $u = 2 x$ .
  2. The derivative of $e^{u}$ is itself.
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $2$
    The result of the chain rule is:
    
    $2 e^{2 x}$
  The result of the chain rule is:
  
  $- 2 e^{2 x} \sin{\left(e^{2 x} \right)}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  /       2/ 2*x\\  2*x
2*\1 + tan \E   //*e

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

Simplify

The second derivative [src]

  /       2/ 2*x\\ /       2*x    / 2*x\\  2*x
4*\1 + tan \E   //*\1 + 2*e   *tan\E   //*e

$$4 \left(2 e^{2 x} \tan{\left(e^{2 x} \right)} + 1\right) \left(\tan^{2}{\left(e^{2 x} \right)} + 1\right) e^{2 x}$$

Simplify

The third derivative [src]

  /       2/ 2*x\\ /      /       2/ 2*x\\  4*x        2/ 2*x\  4*x      2*x    / 2*x\\  2*x
8*\1 + tan \E   //*\1 + 2*\1 + tan \E   //*e    + 4*tan \E   /*e    + 6*e   *tan\E   //*e

$$8 \left(\tan^{2}{\left(e^{2 x} \right)} + 1\right) \left(2 \left(\tan^{2}{\left(e^{2 x} \right)} + 1\right) e^{4 x} + 4 e^{4 x} \tan^{2}{\left(e^{2 x} \right)} + 6 e^{2 x} \tan{\left(e^{2 x} \right)} + 1\right) e^{2 x}$$

Simplify

The graph

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Identical expressions

Derivative of tan(e^(2*x))

The graph:

Piecewise:

The solution