Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of x^2*e^2
Derivative of 3^-x
Derivative of e^(x*(-4))
Derivative of (3x-5)^3
Identical expressions
sin(ln(5x^ eight +8x))
sinus of (ln(5x to the power of 8 plus 8x))
sinus of (ln(5x to the power of eight plus 8x))
sin(ln(5x8+8x))
sinln5x8+8x
sin(ln(5x⁸+8x))
sinln5x^8+8x
Similar expressions
sin(ln(5x^8-8x))

The derivative of the function/ sin(ln(5x^8+8x))

Derivative of sin(ln(5x^8+8x))

The solution

You have entered [src]

   /   /   8      \\
sin\log\5*x  + 8*x//

$$\sin{\left(\log{\left(5 x^{8} + 8 x \right)} \right)}$$

sin(log(5*x^8 + 8*x))

Detail solution

Let $u = \log{\left(5 x^{8} + 8 x \right)}$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(5 x^{8} + 8 x \right)}$ :
1. Let $u = 5 x^{8} + 8 x$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x^{8} + 8 x\right)$ :
  1. Differentiate $5 x^{8} + 8 x$ 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: $x^{8}$ goes to $8 x^{7}$
      So, the result is: $40 x^{7}$
    2. 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: $8$
    The result is: $40 x^{7} + 8$
  The result of the chain rule is:
  
  $\frac{40 x^{7} + 8}{5 x^{8} + 8 x}$
The result of the chain rule is:

$\frac{\left(40 x^{7} + 8\right) \cos{\left(\log{\left(5 x^{8} + 8 x \right)} \right)}}{5 x^{8} + 8 x}$
Now simplify:

$\frac{8 \left(5 x^{7} + 1\right) \cos{\left(\log{\left(x \left(5 x^{7} + 8\right) \right)} \right)}}{x \left(5 x^{7} + 8\right)}$

The answer is:

$\frac{8 \left(5 x^{7} + 1\right) \cos{\left(\log{\left(x \left(5 x^{7} + 8\right) \right)} \right)}}{x \left(5 x^{7} + 8\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/        7\    /   /   8      \\
\8 + 40*x /*cos\log\5*x  + 8*x//
--------------------------------
              8                 
           5*x  + 8*x

$$\frac{\left(40 x^{7} + 8\right) \cos{\left(\log{\left(5 x^{8} + 8 x \right)} \right)}}{5 x^{8} + 8 x}$$

Simplify

The second derivative [src]

  /                                           2                                      2                       \
  |                                 /       7\     /   /  /       7\\\     /       7\     /   /  /       7\\\|
  |    5    /   /  /       7\\\   8*\1 + 5*x / *cos\log\x*\8 + 5*x ///   8*\1 + 5*x / *sin\log\x*\8 + 5*x ///|
8*|35*x *cos\log\x*\8 + 5*x /// - ------------------------------------ - ------------------------------------|
  |                                           2 /       7\                           2 /       7\            |
  \                                          x *\8 + 5*x /                          x *\8 + 5*x /            /
--------------------------------------------------------------------------------------------------------------
                                                          7                                                   
                                                   8 + 5*x

$$\frac{8 \left(35 x^{5} \cos{\left(\log{\left(x \left(5 x^{7} + 8\right) \right)} \right)} - \frac{8 \left(5 x^{7} + 1\right)^{2} \sin{\left(\log{\left(x \left(5 x^{7} + 8\right) \right)} \right)}}{x^{2} \left(5 x^{7} + 8\right)} - \frac{8 \left(5 x^{7} + 1\right)^{2} \cos{\left(\log{\left(x \left(5 x^{7} + 8\right) \right)} \right)}}{x^{2} \left(5 x^{7} + 8\right)}\right)}{5 x^{7} + 8}$$

Simplify

The third derivative [src]

   /                                                                                                                                   3                                       3                       \
   |                                     4 /       7\    /   /  /       7\\\        4 /       7\    /   /  /       7\\\      /       7\     /   /  /       7\\\      /       7\     /   /  /       7\\\|
   |     4    /   /  /       7\\\   420*x *\1 + 5*x /*cos\log\x*\8 + 5*x ///   420*x *\1 + 5*x /*sin\log\x*\8 + 5*x ///   32*\1 + 5*x / *cos\log\x*\8 + 5*x ///   96*\1 + 5*x / *sin\log\x*\8 + 5*x ///|
16*|105*x *cos\log\x*\8 + 5*x /// - ---------------------------------------- - ---------------------------------------- + ------------------------------------- + -------------------------------------|
   |                                                       7                                          7                                            2                                       2           |
   |                                                8 + 5*x                                    8 + 5*x                                 3 /       7\                            3 /       7\            |
   \                                                                                                                                  x *\8 + 5*x /                           x *\8 + 5*x /            /
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                       7                                                                                                
                                                                                                8 + 5*x

$$\frac{16 \left(- \frac{420 x^{4} \left(5 x^{7} + 1\right) \sin{\left(\log{\left(x \left(5 x^{7} + 8\right) \right)} \right)}}{5 x^{7} + 8} - \frac{420 x^{4} \left(5 x^{7} + 1\right) \cos{\left(\log{\left(x \left(5 x^{7} + 8\right) \right)} \right)}}{5 x^{7} + 8} + 105 x^{4} \cos{\left(\log{\left(x \left(5 x^{7} + 8\right) \right)} \right)} + \frac{96 \left(5 x^{7} + 1\right)^{3} \sin{\left(\log{\left(x \left(5 x^{7} + 8\right) \right)} \right)}}{x^{3} \left(5 x^{7} + 8\right)^{2}} + \frac{32 \left(5 x^{7} + 1\right)^{3} \cos{\left(\log{\left(x \left(5 x^{7} + 8\right) \right)} \right)}}{x^{3} \left(5 x^{7} + 8\right)^{2}}\right)}{5 x^{7} + 8}$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of sin(ln(5x^8+8x))

The graph:

Piecewise:

The solution