Mister Exam

Other calculators

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

Derivative of cos(x)*log(x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
cos(x)*log(x)
log⁡(x)cos⁡(x)\log{\left(x \right)} \cos{\left(x \right)} 
cos(x)*log(x)
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⁡(x)f{\left(x \right)} = \cos{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. The derivative of cosine is negative sine:

      ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(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: −log⁡(x)sin⁡(x)+cos⁡(x)x- \log{\left(x \right)} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x}

The answer is:

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

The graph
02468-8-6-4-2-101020-10
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
cos(x)                
------ - log(x)*sin(x)
  x
−log⁡(x)sin⁡(x)+cos⁡(x)x- \log{\left(x \right)} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x}
Simplify
The second derivative [src] 
 /cos(x)                   2*sin(x)\
-|------ + cos(x)*log(x) + --------|
 |   2                        x    |
 \  x                              /
−(log⁡(x)cos⁡(x)+2sin⁡(x)x+cos⁡(x)x2)- (\log{\left(x \right)} \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{\cos{\left(x \right)}}{x^{2}})
Simplify
The third derivative [src] 
                3*cos(x)   2*cos(x)   3*sin(x)
log(x)*sin(x) - -------- + -------- + --------
                   x           3          2   
                              x          x
log⁡(x)sin⁡(x)−3cos⁡(x)x+3sin⁡(x)x2+2cos⁡(x)x3\log{\left(x \right)} \sin{\left(x \right)} - \frac{3 \cos{\left(x \right)}}{x} + \frac{3 \sin{\left(x \right)}}{x^{2}} + \frac{2 \cos{\left(x \right)}}{x^{3}}
Simplify
The graph
Derivative of cos(x)*log(x)
    © Mister Exam – Calculator