Derivative of y=lntg(2x+5)=ln5

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log(tan(2*x + 5))

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

log(tan(2*x + 5))

Detail solution

Let $u = \tan{\left(2 x + 5 \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(2 x + 5 \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(2 x + 5 \right)} = \frac{\sin{\left(2 x + 5 \right)}}{\cos{\left(2 x + 5 \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(2 x + 5 \right)}$ and $g{\left(x \right)} = \cos{\left(2 x + 5 \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 2 x + 5$ .
  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 $\frac{d}{d x} \left(2 x + 5\right)$ :
    1. Differentiate $2 x + 5$ term by term:
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
      2. The derivative of the constant $5$ is zero.
      The result is: $2$
    The result of the chain rule is:
    
    $2 \cos{\left(2 x + 5 \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 2 x + 5$ .
  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 $\frac{d}{d x} \left(2 x + 5\right)$ :
    1. Differentiate $2 x + 5$ term by term:
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
      2. The derivative of the constant $5$ is zero.
      The result is: $2$
    The result of the chain rule is:
    
    $- 2 \sin{\left(2 x + 5 \right)}$
  Now plug in to the quotient rule:
  
  $\frac{2 \sin^{2}{\left(2 x + 5 \right)} + 2 \cos^{2}{\left(2 x + 5 \right)}}{\cos^{2}{\left(2 x + 5 \right)}}$
The result of the chain rule is:

$\frac{2 \sin^{2}{\left(2 x + 5 \right)} + 2 \cos^{2}{\left(2 x + 5 \right)}}{\cos^{2}{\left(2 x + 5 \right)} \tan{\left(2 x + 5 \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                       /                                    2                        \
                       |                 /       2         \      /       2         \|
   /       2         \ |                 \1 + tan (5 + 2*x)/    2*\1 + tan (5 + 2*x)/|
16*\1 + tan (5 + 2*x)/*|2*tan(5 + 2*x) + -------------------- - ---------------------|
                       |                       3                     tan(5 + 2*x)    |
                       \                    tan (5 + 2*x)                            /

$$16 \left(\tan^{2}{\left(2 x + 5 \right)} + 1\right) \left(\frac{\left(\tan^{2}{\left(2 x + 5 \right)} + 1\right)^{2}}{\tan^{3}{\left(2 x + 5 \right)}} - \frac{2 \left(\tan^{2}{\left(2 x + 5 \right)} + 1\right)}{\tan{\left(2 x + 5 \right)}} + 2 \tan{\left(2 x + 5 \right)}\right)$$

Simplify

The graph

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Derivative of y=lntg(2x+5)=ln5

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The solution