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Derivative of atan(2*x)
Identical expressions
x^ three *sin(5x)+ln^2x
x cubed multiply by sinus of (5x) plus ln squared x
x to the power of three multiply by sinus of (5x) plus ln squared x
x3*sin(5x)+ln2x
x3*sin5x+ln2x
x³*sin(5x)+ln²x
x to the power of 3*sin(5x)+ln to the power of 2x
x^3sin(5x)+ln^2x
x3sin(5x)+ln2x
x3sin5x+ln2x
x^3sin5x+ln^2x
Similar expressions
x^3*sin(5x)-ln^2x

The derivative of the function/ x^3*sin(5x)+ln^2x

Derivative of x^3*sin(5x)+ln^2x

The solution

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

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

d / 3               2   \
--\x *sin(5*x) + log (x)/
dx

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

Transform

Detail solution

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

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

2        3            2*log(x)                      2         
-- - 25*x *sin(5*x) - -------- + 6*x*sin(5*x) + 30*x *cos(5*x)
 2                        2                                   
x                        x

$$- 25 x^{3} \sin{\left(5 x \right)} + 30 x^{2} \cos{\left(5 x \right)} + 6 x \sin{\left(5 x \right)} - \frac{2 \log{\left(x \right)}}{x^{2}} + \frac{2}{x^{2}}$$

Simplify

The third derivative [src]

  6                      2                 3            4*log(x)                
- -- + 6*sin(5*x) - 225*x *sin(5*x) - 125*x *cos(5*x) + -------- + 90*x*cos(5*x)
   3                                                        3                   
  x                                                        x

$$- 125 x^{3} \cos{\left(5 x \right)} - 225 x^{2} \sin{\left(5 x \right)} + 90 x \cos{\left(5 x \right)} + 6 \sin{\left(5 x \right)} + \frac{4 \log{\left(x \right)}}{x^{3}} - \frac{6}{x^{3}}$$

Simplify

The graph

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Derivative of x^3*sin(5x)+ln^2x

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