Derivative of f(x)=exp(tan(x)+2x³)

The solution

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             3
 tan(x) + 2*x 
e

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

  /             3\
d | tan(x) + 2*x |
--\e             /
dx

$$\frac{d}{d x} e^{2 x^{3} + \tan{\left(x \right)}}$$

Transform

Detail solution

Let $u = 2 x^{3} + \tan{\left(x \right)}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x^{3} + \tan{\left(x \right)}\right)$ :
1. Differentiate $2 x^{3} + \tan{\left(x \right)}$ term by term:
  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:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  3. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
    So, the result is: $6 x^{2}$
  The result is: $6 x^{2} + \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result of the chain rule is:

$\left(6 x^{2} + \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) e^{2 x^{3} + \tan{\left(x \right)}}$
Now simplify:

$\left(6 x^{2} + \frac{1}{\cos^{2}{\left(x \right)}}\right) e^{2 x^{3} + \tan{\left(x \right)}}$

The answer is:

$\left(6 x^{2} + \frac{1}{\cos^{2}{\left(x \right)}}\right) e^{2 x^{3} + \tan{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                                  3
/       2         2\  tan(x) + 2*x 
\1 + tan (x) + 6*x /*e

$$\left(6 x^{2} + \tan^{2}{\left(x \right)} + 1\right) e^{2 x^{3} + \tan{\left(x \right)}}$$

Simplify

The second derivative [src]

/                    2                                \     3         
|/       2         2\             /       2   \       |  2*x  + tan(x)
\\1 + tan (x) + 6*x /  + 12*x + 2*\1 + tan (x)/*tan(x)/*e

$$\left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(6 x^{2} + \tan^{2}{\left(x \right)} + 1\right)^{2} + 12 x\right) e^{2 x^{3} + \tan{\left(x \right)}}$$

Simplify

The third derivative [src]

/                         3                  2                                                                                \     3         
|     /       2         2\      /       2   \         2    /       2   \     /      /       2   \       \ /       2         2\|  2*x  + tan(x)
\12 + \1 + tan (x) + 6*x /  + 2*\1 + tan (x)/  + 4*tan (x)*\1 + tan (x)/ + 6*\6*x + \1 + tan (x)/*tan(x)/*\1 + tan (x) + 6*x //*e

$$\left(4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \left(6 x^{2} + \tan^{2}{\left(x \right)} + 1\right)^{3} + 6 \cdot \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 6 x\right) \left(6 x^{2} + \tan^{2}{\left(x \right)} + 1\right) + 2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 12\right) e^{2 x^{3} + \tan{\left(x \right)}}$$

Simplify

The graph

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Derivative of f(x)=exp(tan(x)+2x³)

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The solution