Derivative of y=sin(ln5x²+2x)

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   /   2           \
sin\log (5*x) + 2*x/

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

sin(log(5*x)^2 + 2*x)

Detail solution

Let $u = 2 x + \log{\left(5 x \right)}^{2}$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + \log{\left(5 x \right)}^{2}\right)$ :
1. Differentiate $2 x + \log{\left(5 x \right)}^{2}$ term by term:
  1. Let $u = \log{\left(5 x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(5 x \right)}$ :
    1. Let $u = 5 x$ .
    2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$ :
      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: $5$
      The result of the chain rule is:
      
      $\frac{1}{x}$
    The result of the chain rule is:
    
    $\frac{2 \log{\left(5 x \right)}}{x}$
  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: $2$
  The result is: $2 + \frac{2 \log{\left(5 x \right)}}{x}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/    2*log(5*x)\    /   2           \
|2 + ----------|*cos\log (5*x) + 2*x/
\        x     /

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

Simplify

The second derivative [src]

   /                2                                           /   2           \\
   |  /    log(5*x)\     /   2           \   (-1 + log(5*x))*cos\log (5*x) + 2*x/|
-2*|2*|1 + --------| *sin\log (5*x) + 2*x/ + ------------------------------------|
   |  \       x    /                                           2                 |
   \                                                          x                  /

$$- 2 \left(2 \left(1 + \frac{\log{\left(5 x \right)}}{x}\right)^{2} \sin{\left(2 x + \log{\left(5 x \right)}^{2} \right)} + \frac{\left(\log{\left(5 x \right)} - 1\right) \cos{\left(2 x + \log{\left(5 x \right)}^{2} \right)}}{x^{2}}\right)$$

Simplify

The third derivative [src]

  /                                                                                      /    log(5*x)\                    /   2           \\
  |                  3                                             /   2           \   6*|1 + --------|*(-1 + log(5*x))*sin\log (5*x) + 2*x/|
  |    /    log(5*x)\     /   2           \   (-3 + 2*log(5*x))*cos\log (5*x) + 2*x/     \       x    /                                     |
2*|- 4*|1 + --------| *cos\log (5*x) + 2*x/ + -------------------------------------- + -----------------------------------------------------|
  |    \       x    /                                            3                                                2                         |
  \                                                             x                                                x                          /

$$2 \left(- 4 \left(1 + \frac{\log{\left(5 x \right)}}{x}\right)^{3} \cos{\left(2 x + \log{\left(5 x \right)}^{2} \right)} + \frac{6 \left(1 + \frac{\log{\left(5 x \right)}}{x}\right) \left(\log{\left(5 x \right)} - 1\right) \sin{\left(2 x + \log{\left(5 x \right)}^{2} \right)}}{x^{2}} + \frac{\left(2 \log{\left(5 x \right)} - 3\right) \cos{\left(2 x + \log{\left(5 x \right)}^{2} \right)}}{x^{3}}\right)$$

Simplify

The graph

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Derivative of y=sin(ln5x²+2x)

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The solution