Derivative of y=tan²x²+5x-2

The solution

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        / 2\          
        \2 /          
(tan(x))     + 5*x - 2

$$5 x + \tan^{2^{2}}{\left(x \right)} - 2$$

  /        / 2\          \
d |        \2 /          |
--\(tan(x))     + 5*x - 2/
dx

$$\frac{d}{d x} \left(5 x + \tan^{2^{2}}{\left(x \right)} - 2\right)$$

Transform

Detail solution

Differentiate $5 x + \tan^{2^{2}}{\left(x \right)} - 2$ term by term:
1. Let $u = \tan{\left(x \right)}$ .
2. Apply the power rule: $u^{2^{2}}$ goes to $\frac{4 u^{2^{2}}}{u}$
3. 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:
  
  $\frac{4 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2^{2}}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}}$
4. 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: $5$
5. The derivative of the constant $\left(-1\right) 2$ is zero.
The result is: $\frac{4 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2^{2}}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}} + 5$
Now simplify:

$\frac{4 \sin^{3}{\left(x \right)}}{\cos^{5}{\left(x \right)}} + 5$

The answer is:

$\frac{4 \sin^{3}{\left(x \right)}}{\cos^{5}{\left(x \right)}} + 5$

The graph

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Plot the graph f'(x)

The first derivative [src]

            / 2\                
            \2 / /         2   \
    (tan(x))    *\4 + 4*tan (x)/
5 + ----------------------------
               tan(x)

$$\frac{\left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{2^{2}}{\left(x \right)}}{\tan{\left(x \right)}} + 5$$

Simplify

The second derivative [src]

     2    /       2   \ /         2   \
4*tan (x)*\1 + tan (x)/*\3 + 5*tan (x)/

$$4 \left(\tan^{2}{\left(x \right)} + 1\right) \left(5 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)}$$

Simplify

The third derivative [src]

                /                           2                           \       
  /       2   \ |     4        /       2   \          2    /       2   \|       
8*\1 + tan (x)/*\2*tan (x) + 3*\1 + tan (x)/  + 10*tan (x)*\1 + tan (x)//*tan(x)

$$8 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \tan^{4}{\left(x \right)} + 10 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}\right) \tan{\left(x \right)}$$

Simplify

The graph

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Derivative of y=tan²x²+5x-2

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