Mister Exam

Derivative of cosx-log5x

Function f() - derivative -N order at the point
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The graph:

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Piecewise:

The solution

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cos(x) - log(5*x)
log(5x)+cos(x)- \log{\left(5 x \right)} + \cos{\left(x \right)}
cos(x) - log(5*x)
Detail solution
  1. Differentiate log(5x)+cos(x)- \log{\left(5 x \right)} + \cos{\left(x \right)} term by term:

    1. The derivative of cosine is negative sine:

      ddxcos(x)=sin(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

    2. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Let u=5xu = 5 x.

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

      3. Then, apply the chain rule. Multiply by ddx5x\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: xx goes to 11

          So, the result is: 55

        The result of the chain rule is:

        1x\frac{1}{x}

      So, the result is: 1x- \frac{1}{x}

    The result is: sin(x)1x- \sin{\left(x \right)} - \frac{1}{x}


The answer is:

sin(x)1x- \sin{\left(x \right)} - \frac{1}{x}

The graph
02468-8-6-4-2-1010-2020
The first derivative [src]
  1         
- - - sin(x)
  x         
sin(x)1x- \sin{\left(x \right)} - \frac{1}{x}
The second derivative [src]
1          
-- - cos(x)
 2         
x          
cos(x)+1x2- \cos{\left(x \right)} + \frac{1}{x^{2}}
The third derivative [src]
  2          
- -- + sin(x)
   3         
  x          
sin(x)2x3\sin{\left(x \right)} - \frac{2}{x^{3}}
The graph
Derivative of cosx-log5x