Derivative of y=ln4x*cos5x

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log(4*x)*cos(5*x)

$$\log{\left(4 x \right)} \cos{\left(5 x \right)}$$

log(4*x)*cos(5*x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \log{\left(4 x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 4 x$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 4 x$ :
  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: $4$
  The result of the chain rule is:
  
  $\frac{1}{x}$
$g{\left(x \right)} = \cos{\left(5 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 5 x$ .
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 $\frac{d}{d x} 5 x$ :
  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: $5$
  The result of the chain rule is:
  
  $- 5 \sin{\left(5 x \right)}$
The result is: $- 5 \log{\left(4 x \right)} \sin{\left(5 x \right)} + \frac{\cos{\left(5 x \right)}}{x}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

cos(5*x)                      
-------- - 5*log(4*x)*sin(5*x)
   x

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

Simplify

The second derivative [src]

 /cos(5*x)   10*sin(5*x)                       \
-|-------- + ----------- + 25*cos(5*x)*log(4*x)|
 |    2           x                            |
 \   x                                         /

$$- (25 \log{\left(4 x \right)} \cos{\left(5 x \right)} + \frac{10 \sin{\left(5 x \right)}}{x} + \frac{\cos{\left(5 x \right)}}{x^{2}})$$

Simplify

The third derivative [src]

  75*cos(5*x)   2*cos(5*x)   15*sin(5*x)                        
- ----------- + ---------- + ----------- + 125*log(4*x)*sin(5*x)
       x             3             2                            
                    x             x

$$125 \log{\left(4 x \right)} \sin{\left(5 x \right)} - \frac{75 \cos{\left(5 x \right)}}{x} + \frac{15 \sin{\left(5 x \right)}}{x^{2}} + \frac{2 \cos{\left(5 x \right)}}{x^{3}}$$

Simplify

The graph

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Derivative of y=ln4x*cos5x

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The solution