Mister Exam

Derivative of cos(3x)*log(x)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
cos(3*x)*log(x)
log(x)cos(3x)\log{\left(x \right)} \cos{\left(3 x \right)}
d                  
--(cos(3*x)*log(x))
dx                 
ddxlog(x)cos(3x)\frac{d}{d x} \log{\left(x \right)} \cos{\left(3 x \right)}
Detail solution
  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=cos(3x)f{\left(x \right)} = \cos{\left(3 x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Let u=3xu = 3 x.

    2. The derivative of cosine is negative sine:

      dducos(u)=sin(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

    3. Then, apply the chain rule. Multiply by ddx3x\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: xx goes to 11

        So, the result is: 33

      The result of the chain rule is:

      3sin(3x)- 3 \sin{\left(3 x \right)}

    g(x)=log(x)g{\left(x \right)} = \log{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. The derivative of log(x)\log{\left(x \right)} is 1x\frac{1}{x}.

    The result is: 3log(x)sin(3x)+cos(3x)x- 3 \log{\left(x \right)} \sin{\left(3 x \right)} + \frac{\cos{\left(3 x \right)}}{x}


The answer is:

3log(x)sin(3x)+cos(3x)x- 3 \log{\left(x \right)} \sin{\left(3 x \right)} + \frac{\cos{\left(3 x \right)}}{x}

The graph
02468-8-6-4-2-1010-2020
The first derivative [src]
cos(3*x)                    
-------- - 3*log(x)*sin(3*x)
   x                        
3log(x)sin(3x)+cos(3x)x- 3 \log{\left(x \right)} \sin{\left(3 x \right)} + \frac{\cos{\left(3 x \right)}}{x}
The second derivative [src]
 /cos(3*x)   6*sin(3*x)                    \
-|-------- + ---------- + 9*cos(3*x)*log(x)|
 |    2          x                         |
 \   x                                     /
(9log(x)cos(3x)+6sin(3x)x+cos(3x)x2)- (9 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{6 \sin{\left(3 x \right)}}{x} + \frac{\cos{\left(3 x \right)}}{x^{2}})
The third derivative [src]
  27*cos(3*x)   2*cos(3*x)   9*sin(3*x)                     
- ----------- + ---------- + ---------- + 27*log(x)*sin(3*x)
       x             3            2                         
                    x            x                          
27log(x)sin(3x)27cos(3x)x+9sin(3x)x2+2cos(3x)x327 \log{\left(x \right)} \sin{\left(3 x \right)} - \frac{27 \cos{\left(3 x \right)}}{x} + \frac{9 \sin{\left(3 x \right)}}{x^{2}} + \frac{2 \cos{\left(3 x \right)}}{x^{3}}
The graph
Derivative of cos(3x)*log(x)