Derivative of (sin8x*ln(2x+5))

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sin(8*x)*log(2*x + 5)

$$\log{\left(2 x + 5 \right)} \sin{\left(8 x \right)}$$

sin(8*x)*log(2*x + 5)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sin{\left(8 x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 8 x$ .
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 $\frac{d}{d x} 8 x$ :
  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: $8$
  The result of the chain rule is:
  
  $8 \cos{\left(8 x \right)}$
$g{\left(x \right)} = \log{\left(2 x + 5 \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 2 x + 5$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
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.
      1. 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:
  
  $\frac{2}{2 x + 5}$
The result is: $8 \log{\left(2 x + 5 \right)} \cos{\left(8 x \right)} + \frac{2 \sin{\left(8 x \right)}}{2 x + 5}$
Now simplify:

$\frac{2 \left(4 \left(2 x + 5\right) \log{\left(2 x + 5 \right)} \cos{\left(8 x \right)} + \sin{\left(8 x \right)}\right)}{2 x + 5}$

The answer is:

$\frac{2 \left(4 \left(2 x + 5\right) \log{\left(2 x + 5 \right)} \cos{\left(8 x \right)} + \sin{\left(8 x \right)}\right)}{2 x + 5}$

The graph

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The first derivative [src]

2*sin(8*x)                          
---------- + 8*cos(8*x)*log(2*x + 5)
 2*x + 5

$$8 \log{\left(2 x + 5 \right)} \cos{\left(8 x \right)} + \frac{2 \sin{\left(8 x \right)}}{2 x + 5}$$

Simplify

The second derivative [src]

  /   sin(8*x)                               8*cos(8*x)\
4*|- ---------- - 16*log(5 + 2*x)*sin(8*x) + ----------|
  |           2                               5 + 2*x  |
  \  (5 + 2*x)                                         /

$$4 \left(- 16 \log{\left(2 x + 5 \right)} \sin{\left(8 x \right)} + \frac{8 \cos{\left(8 x \right)}}{2 x + 5} - \frac{\sin{\left(8 x \right)}}{\left(2 x + 5\right)^{2}}\right)$$

Simplify

The third derivative [src]

   / sin(8*x)                               24*sin(8*x)   6*cos(8*x)\
16*|---------- - 32*cos(8*x)*log(5 + 2*x) - ----------- - ----------|
   |         3                                5 + 2*x              2|
   \(5 + 2*x)                                             (5 + 2*x) /

$$16 \left(- 32 \log{\left(2 x + 5 \right)} \cos{\left(8 x \right)} - \frac{24 \sin{\left(8 x \right)}}{2 x + 5} - \frac{6 \cos{\left(8 x \right)}}{\left(2 x + 5\right)^{2}} + \frac{\sin{\left(8 x \right)}}{\left(2 x + 5\right)^{3}}\right)$$

Simplify

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Derivative of (sin8x*ln(2x+5))

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