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Identical expressions
ln(3x)*cos(x^ two + five)
ln(3x) multiply by co sinus of e of (x squared plus 5)
ln(3x) multiply by co sinus of e of (x to the power of two plus five)
ln(3x)*cos(x2+5)
ln3x*cosx2+5
ln(3x)*cos(x²+5)
ln(3x)*cos(x to the power of 2+5)
ln(3x)cos(x^2+5)
ln(3x)cos(x2+5)
ln3xcosx2+5
ln3xcosx^2+5
Similar expressions
ln(3x)*cos(x^2-5)

The derivative of the function/ ln(3x)*cos(x^2+5)

Derivative of ln(3x)*cos(x^2+5)

The solution

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            / 2    \
log(3*x)*cos\x  + 5/

$$\log{\left(3 x \right)} \cos{\left(x^{2} + 5 \right)}$$

log(3*x)*cos(x^2 + 5)

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   / 2    \                           
cos\x  + 5/                   / 2    \
----------- - 2*x*log(3*x)*sin\x  + 5/
     x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of ln(3x)*cos(x^2+5)

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The solution