Derivative of y=ln*sin^4x

The solution

You have entered [src]

          4   
log(x)*sin (x)

$$\log{\left(x \right)} \sin^{4}{\left(x \right)}$$

d /          4   \
--\log(x)*sin (x)/
dx

$$\frac{d}{d x} \log{\left(x \right)} \sin^{4}{\left(x \right)}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
$g{\left(x \right)} = \sin^{4}{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$
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:
  
  $4 \sin^{3}{\left(x \right)} \cos{\left(x \right)}$
The result is: $4 \log{\left(x \right)} \sin^{3}{\left(x \right)} \cos{\left(x \right)} + \frac{\sin^{4}{\left(x \right)}}{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   4                             
sin (x)        3                 
------- + 4*sin (x)*cos(x)*log(x)
   x

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

Simplify

The second derivative [src]

        /     2                                                      \
   2    |  sin (x)     /   2           2   \          8*cos(x)*sin(x)|
sin (x)*|- ------- - 4*\sin (x) - 3*cos (x)/*log(x) + ---------------|
        |      2                                             x       |
        \     x                                                      /

$$\left(- 4 \left(\sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right) \log{\left(x \right)} + \frac{8 \sin{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right) \sin^{2}{\left(x \right)}$$

Simplify

The third derivative [src]

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

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

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Derivative of y=ln*sin^4x

The graph:

Piecewise:

The solution