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Identical expressions
sin^5ln(3x^ two +x)
sinus of to the power of 5ln(3x squared plus x)
sinus of to the power of 5ln(3x to the power of two plus x)
sin5ln(3x2+x)
sin5ln3x2+x
sin⁵ln(3x²+x)
sin to the power of 5ln(3x to the power of 2+x)
sin^5ln3x^2+x
Similar expressions
sin^5ln(3x^2-x)

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sin^{5}{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $u^{5}$ goes to $5 u^{4}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $5 \sin^{4}{\left(x \right)} \cos{\left(x \right)}$
$g{\left(x \right)} = \log{\left(3 x^{2} + x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 3 x^{2} + x$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x^{2} + x\right)$ :
  1. Differentiate $3 x^{2} + x$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{2}$ goes to $2 x$
      So, the result is: $6 x$
    2. Apply the power rule: $x$ goes to $1$
    The result is: $6 x + 1$
  The result of the chain rule is:
  
  $\frac{6 x + 1}{3 x^{2} + x}$
The result is: $\frac{\left(6 x + 1\right) \sin^{5}{\left(x \right)}}{3 x^{2} + x} + 5 \log{\left(3 x^{2} + x \right)} \sin^{4}{\left(x \right)} \cos{\left(x \right)}$
Now simplify:

$\frac{\left(5 x \left(3 x + 1\right) \log{\left(x \left(3 x + 1\right) \right)} \cos{\left(x \right)} + \left(6 x + 1\right) \sin{\left(x \right)}\right) \sin^{4}{\left(x \right)}}{x \left(3 x + 1\right)}$

The answer is:

$\frac{\left(5 x \left(3 x + 1\right) \log{\left(x \left(3 x + 1\right) \right)} \cos{\left(x \right)} + \left(6 x + 1\right) \sin{\left(x \right)}\right) \sin^{4}{\left(x \right)}}{x \left(3 x + 1\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   5                                              
sin (x)*(1 + 6*x)        4              /   2    \
----------------- + 5*sin (x)*cos(x)*log\3*x  + x/
        2                                         
     3*x  + x

$$\frac{\left(6 x + 1\right) \sin^{5}{\left(x \right)}}{3 x^{2} + x} + 5 \log{\left(3 x^{2} + x \right)} \sin^{4}{\left(x \right)} \cos{\left(x \right)}$$

Simplify

The second derivative [src]

        /                                                     /              2\                             \
        |                                                2    |     (1 + 6*x) |                             |
        |                                             sin (x)*|6 - -----------|                             |
   3    |    /   2           2   \                            \    x*(1 + 3*x)/   10*(1 + 6*x)*cos(x)*sin(x)|
sin (x)*|- 5*\sin (x) - 4*cos (x)/*log(x*(1 + 3*x)) + ------------------------- + --------------------------|
        \                                                    x*(1 + 3*x)                 x*(1 + 3*x)        /

$$\left(- 5 \left(\sin^{2}{\left(x \right)} - 4 \cos^{2}{\left(x \right)}\right) \log{\left(x \left(3 x + 1\right) \right)} + \frac{\left(6 - \frac{\left(6 x + 1\right)^{2}}{x \left(3 x + 1\right)}\right) \sin^{2}{\left(x \right)}}{x \left(3 x + 1\right)} + \frac{10 \left(6 x + 1\right) \sin{\left(x \right)} \cos{\left(x \right)}}{x \left(3 x + 1\right)}\right) \sin^{3}{\left(x \right)}$$

Simplify

The third derivative [src]

        /                                                                                                                          /              2\              /              2\       \
        |                                                                                                           3              |     (1 + 6*x) |         2    |     (1 + 6*x) |       |
        |                                                                       /   2           2   \          2*sin (x)*(1 + 6*x)*|9 - -----------|   15*sin (x)*|6 - -----------|*cos(x)|
   2    |    /        2            2   \                           15*(1 + 6*x)*\sin (x) - 4*cos (x)/*sin(x)                       \    x*(1 + 3*x)/              \    x*(1 + 3*x)/       |
sin (x)*|- 5*\- 12*cos (x) + 13*sin (x)/*cos(x)*log(x*(1 + 3*x)) - ----------------------------------------- - ------------------------------------- + -----------------------------------|
        |                                                                         x*(1 + 3*x)                               2          2                           x*(1 + 3*x)            |
        \                                                                                                                  x *(1 + 3*x)                                                   /

$$\left(- 5 \left(13 \sin^{2}{\left(x \right)} - 12 \cos^{2}{\left(x \right)}\right) \log{\left(x \left(3 x + 1\right) \right)} \cos{\left(x \right)} + \frac{15 \left(6 - \frac{\left(6 x + 1\right)^{2}}{x \left(3 x + 1\right)}\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x \left(3 x + 1\right)} - \frac{15 \left(6 x + 1\right) \left(\sin^{2}{\left(x \right)} - 4 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{x \left(3 x + 1\right)} - \frac{2 \left(9 - \frac{\left(6 x + 1\right)^{2}}{x \left(3 x + 1\right)}\right) \left(6 x + 1\right) \sin^{3}{\left(x \right)}}{x^{2} \left(3 x + 1\right)^{2}}\right) \sin^{2}{\left(x \right)}$$

Simplify

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

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The solution