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Derivative of arcctg^25xln(x-4)

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    2                
acot (5)*x*log(x - 4)

$$x \operatorname{acot}^{2}{\left(5 \right)} \log{\left(x - 4 \right)}$$

(acot(5)^2*x)*log(x - 4)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x \operatorname{acot}^{2}{\left(5 \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
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: $\operatorname{acot}^{2}{\left(5 \right)}$
$g{\left(x \right)} = \log{\left(x - 4 \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x - 4$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 4\right)$ :
  1. Differentiate $x - 4$ term by term:
    1. Apply the power rule: $x$ goes to $1$
    2. The derivative of the constant $-4$ is zero.
    The result is: $1$
  The result of the chain rule is:
  
  $\frac{1}{x - 4}$
The result is: $\frac{x \operatorname{acot}^{2}{\left(5 \right)}}{x - 4} + \log{\left(x - 4 \right)} \operatorname{acot}^{2}{\left(5 \right)}$
Now simplify:

$\frac{\left(x + \left(x - 4\right) \log{\left(x - 4 \right)}\right) \operatorname{acot}^{2}{\left(5 \right)}}{x - 4}$

The answer is:

$\frac{\left(x + \left(x - 4\right) \log{\left(x - 4 \right)}\right) \operatorname{acot}^{2}{\left(5 \right)}}{x - 4}$

The first derivative [src]

                            2   
    2                 x*acot (5)
acot (5)*log(x - 4) + ----------
                        x - 4

$$\frac{x \operatorname{acot}^{2}{\left(5 \right)}}{x - 4} + \log{\left(x - 4 \right)} \operatorname{acot}^{2}{\left(5 \right)}$$

Simplify

The second derivative [src]

    2    /      x   \
acot (5)*|2 - ------|
         \    -4 + x/
---------------------
        -4 + x

$$\frac{\left(- \frac{x}{x - 4} + 2\right) \operatorname{acot}^{2}{\left(5 \right)}}{x - 4}$$

Simplify

The third derivative [src]

    2    /      2*x  \
acot (5)*|-3 + ------|
         \     -4 + x/
----------------------
              2       
      (-4 + x)

$$\frac{\left(\frac{2 x}{x - 4} - 3\right) \operatorname{acot}^{2}{\left(5 \right)}}{\left(x - 4\right)^{2}}$$

Simplify

Derivative of arcctg^2*5x*ln(x-4)