Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of x^-(4/5)
Derivative of ln(x^2-2x)
Derivative of 2*x-x^2
Derivative of 2^x*x^2
Identical expressions
sqrt(one - eight sinx/8)
square root of (1 minus 8 sinus of x divide by 8)
square root of (one minus eight sinus of x divide by 8)
√(1-8sinx/8)
sqrt1-8sinx/8
sqrt(1-8sinx divide by 8)
Similar expressions
sqrt(1+8sinx/8)

The derivative of the function/ sqrt(1-8sinx/8)

Derivative of sqrt(1-8sinx/8)

The solution

You have entered [src]

    ______________
   /     8*sin(x) 
  /  1 - -------- 
\/          8

$$\sqrt{- \frac{8 \sin{\left(x \right)}}{8} + 1}$$

sqrt(1 - 8*sin(x)/8)

Detail solution

Let $u = - \frac{8 \sin{\left(x \right)}}{8} + 1$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- \frac{8 \sin{\left(x \right)}}{8} + 1\right)$ :
1. Differentiate $- \frac{8 \sin{\left(x \right)}}{8} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. The derivative of sine is cosine:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      So, the result is: $8 \cos{\left(x \right)}$
    So, the result is: $- \cos{\left(x \right)}$
  The result is: $- \cos{\left(x \right)}$
The result of the chain rule is:

$- \frac{\cos{\left(x \right)}}{2 \sqrt{- \frac{8 \sin{\left(x \right)}}{8} + 1}}$
Now simplify:

$- \frac{\cos{\left(x \right)}}{2 \sqrt{1 - \sin{\left(x \right)}}}$

The answer is:

$- \frac{\cos{\left(x \right)}}{2 \sqrt{1 - \sin{\left(x \right)}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      -cos(x)       
--------------------
      ______________
     /     8*sin(x) 
2*  /  1 - -------- 
  \/          8

$$- \frac{\cos{\left(x \right)}}{2 \sqrt{- \frac{8 \sin{\left(x \right)}}{8} + 1}}$$

Simplify

The second derivative [src]

               2     
            cos (x)  
2*sin(x) - ----------
           1 - sin(x)
---------------------
       ____________  
   4*\/ 1 - sin(x)

$$\frac{2 \sin{\left(x \right)} - \frac{\cos^{2}{\left(x \right)}}{1 - \sin{\left(x \right)}}}{4 \sqrt{1 - \sin{\left(x \right)}}}$$

Simplify

The third derivative [src]

/           2                  \       
|      3*cos (x)      6*sin(x) |       
|4 - ------------- + ----------|*cos(x)
|                2   1 - sin(x)|       
\    (1 - sin(x))              /       
---------------------------------------
                ____________           
            8*\/ 1 - sin(x)

$$\frac{\left(4 + \frac{6 \sin{\left(x \right)}}{1 - \sin{\left(x \right)}} - \frac{3 \cos^{2}{\left(x \right)}}{\left(1 - \sin{\left(x \right)}\right)^{2}}\right) \cos{\left(x \right)}}{8 \sqrt{1 - \sin{\left(x \right)}}}$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of sqrt(1-8sinx/8)

The graph:

Piecewise:

The solution