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Derivative of (x^2-4)^3
Derivative of 1/z
Derivative of (x+4)^1/2
Derivative of x^3-9x^2+7
Identical expressions
cos(9x)*(log(x)/log(seven))
co sinus of e of (9x) multiply by ( logarithm of (x) divide by logarithm of (7))
co sinus of e of (9x) multiply by ( logarithm of (x) divide by logarithm of (seven))
cos(9x)(log(x)/log(7))
cos9xlogx/log7
cos(9x)*(log(x) divide by log(7))

The derivative of the function/ cos(9x)*(log(x)/log(7))

Derivative of cos(9x)*(log(x)/log(7))

The solution

You have entered [src]

         log(x)
cos(9*x)*------
         log(7)

$$\frac{\log{\left(x \right)}}{\log{\left(7 \right)}} \cos{\left(9 x \right)}$$

cos(9*x)*(log(x)/log(7))

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \log{\left(x \right)} \cos{\left(9 x \right)}$ and $g{\left(x \right)} = \log{\left(7 \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \cos{\left(9 x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 9 x$ .
  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 $\frac{d}{d x} 9 x$ :
    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: $9$
    The result of the chain rule is:
    
    $- 9 \sin{\left(9 x \right)}$
  $g{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result is: $- 9 \log{\left(x \right)} \sin{\left(9 x \right)} + \frac{\cos{\left(9 x \right)}}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of the constant $\log{\left(7 \right)}$ is zero.
Now plug in to the quotient rule:

$\frac{- 9 \log{\left(x \right)} \sin{\left(9 x \right)} + \frac{\cos{\left(9 x \right)}}{x}}{\log{\left(7 \right)}}$
Now simplify:

$\frac{- 9 x \log{\left(x \right)} \sin{\left(9 x \right)} + \cos{\left(9 x \right)}}{x \log{\left(7 \right)}}$

The answer is:

$\frac{- 9 x \log{\left(x \right)} \sin{\left(9 x \right)} + \cos{\left(9 x \right)}}{x \log{\left(7 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

cos(9*x)   9*log(x)*sin(9*x)
-------- - -----------------
x*log(7)         log(7)

$$- \frac{9 \log{\left(x \right)} \sin{\left(9 x \right)}}{\log{\left(7 \right)}} + \frac{\cos{\left(9 x \right)}}{x \log{\left(7 \right)}}$$

Simplify

The second derivative [src]

 /cos(9*x)   18*sin(9*x)                     \ 
-|-------- + ----------- + 81*cos(9*x)*log(x)| 
 |    2           x                          | 
 \   x                                       / 
-----------------------------------------------
                     log(7)

$$- \frac{81 \log{\left(x \right)} \cos{\left(9 x \right)} + \frac{18 \sin{\left(9 x \right)}}{x} + \frac{\cos{\left(9 x \right)}}{x^{2}}}{\log{\left(7 \right)}}$$

Simplify

The third derivative [src]

  243*cos(9*x)   2*cos(9*x)   27*sin(9*x)                      
- ------------ + ---------- + ----------- + 729*log(x)*sin(9*x)
       x              3             2                          
                     x             x                           
---------------------------------------------------------------
                             log(7)

$$\frac{729 \log{\left(x \right)} \sin{\left(9 x \right)} - \frac{243 \cos{\left(9 x \right)}}{x} + \frac{27 \sin{\left(9 x \right)}}{x^{2}} + \frac{2 \cos{\left(9 x \right)}}{x^{3}}}{\log{\left(7 \right)}}$$

Simplify

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Derivative of cos(9x)*(log(x)/log(7))

The graph:

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The solution