Derivative of sinx*log5x

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sin(x)*log(5*x)

$$\log{\left(5 x \right)} \sin{\left(x \right)}$$

sin(x)*log(5*x)

Detail solution

Apply the product rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  sin(x)                     2*cos(x)
- ------ - log(5*x)*sin(x) + --------
     2                          x    
    x

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

Simplify

The third derivative [src]

                   3*sin(x)   3*cos(x)   2*sin(x)
-cos(x)*log(5*x) - -------- - -------- + --------
                      x           2          3   
                                 x          x

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

Simplify

The graph

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Derivative of sinx*log5x

The graph:

Piecewise:

The solution