Derivative of y=ln^4sinx

The solution

You have entered [src]

   4          
log (x)*sin(x)

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

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

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

Transform

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Derivative of y=ln^4sinx

The graph:

Piecewise:

The solution