Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of x^2/4
Derivative of e^x-1
Derivative of 1/2*x
Derivative of (-1)/x^3
Identical expressions
one /sqrt(one +(x- one)^ four)
1 divide by square root of (1 plus (x minus 1) to the power of 4)
one divide by square root of (one plus (x minus one) to the power of four)
1/√(1+(x-1)^4)
1/sqrt(1+(x-1)4)
1/sqrt1+x-14
1/sqrt(1+(x-1)⁴)
1/sqrt1+x-1^4
1 divide by sqrt(1+(x-1)^4)
Similar expressions
1/sqrt(1-(x-1)^4)
1/sqrt(1+(x+1)^4)

The derivative of the function/ 1/sqrt(1+(x-1)^4)

Derivative of 1/sqrt(1+(x-1)^4)

The solution

You have entered [src]

        1        
-----------------
   ______________
  /            4 
\/  1 + (x - 1)

$$\frac{1}{\sqrt{\left(x - 1\right)^{4} + 1}}$$

1/(sqrt(1 + (x - 1)^4))

Detail solution

Let $u = \sqrt{\left(x - 1\right)^{4} + 1}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{\left(x - 1\right)^{4} + 1}$ :
1. Let $u = \left(x - 1\right)^{4} + 1$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(x - 1\right)^{4} + 1\right)$ :
  1. Differentiate $\left(x - 1\right)^{4} + 1$ term by term:
    1. The derivative of the constant $1$ is zero.
    2. Let $u = x - 1$ .
    3. Apply the power rule: $u^{4}$ goes to $4 u^{3}$
    4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 1\right)$ :
      1. Differentiate $x - 1$ term by term:
        
        Apply the power rule: $x$ goes to $1$
        
        The derivative of the constant $-1$ is zero.
        
        The result is: $1$
      The result of the chain rule is:
      
      $4 \left(x - 1\right)^{3}$
    The result is: $4 \left(x - 1\right)^{3}$
  The result of the chain rule is:
  
  $\frac{2 \left(x - 1\right)^{3}}{\sqrt{\left(x - 1\right)^{4} + 1}}$
The result of the chain rule is:

$- \frac{2 \left(x - 1\right)^{3}}{\left(\left(x - 1\right)^{4} + 1\right)^{\frac{3}{2}}}$
Now simplify:

$- \frac{2 \left(x - 1\right)^{3}}{\left(\left(x - 1\right)^{4} + 1\right)^{\frac{3}{2}}}$

The answer is:

$- \frac{2 \left(x - 1\right)^{3}}{\left(\left(x - 1\right)^{4} + 1\right)^{\frac{3}{2}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                    3           
          -2*(x - 1)            
--------------------------------
                  ______________
/           4\   /            4 
\1 + (x - 1) /*\/  1 + (x - 1)

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

Simplify

The second derivative [src]

            /                4 \
          2 |      2*(-1 + x)  |
6*(-1 + x) *|-1 + -------------|
            |                 4|
            \     1 + (-1 + x) /
--------------------------------
                      3/2       
       /            4\          
       \1 + (-1 + x) /

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

Simplify

The third derivative [src]

            /                  8                4 \
            |       10*(-1 + x)       9*(-1 + x)  |
12*(-1 + x)*|-1 - ---------------- + -------------|
            |                    2               4|
            |     /            4\    1 + (-1 + x) |
            \     \1 + (-1 + x) /                 /
---------------------------------------------------
                                3/2                
                 /            4\                   
                 \1 + (-1 + x) /

$$\frac{12 \left(x - 1\right) \left(- \frac{10 \left(x - 1\right)^{8}}{\left(\left(x - 1\right)^{4} + 1\right)^{2}} + \frac{9 \left(x - 1\right)^{4}}{\left(x - 1\right)^{4} + 1} - 1\right)}{\left(\left(x - 1\right)^{4} + 1\right)^{\frac{3}{2}}}$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of 1/sqrt(1+(x-1)^4)

The graph:

Piecewise:

The solution