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Derivative of:
Derivative of 8^x
Derivative of 2*sinh(x)*cosh(x)
Derivative of x^2*asinh(x)*acosh(x)
Derivative of (x^2-4)/x
Identical expressions
sqrt(5x+ one)^ eight
square root of (5x plus 1) to the power of 8
square root of (5x plus one) to the power of eight
√(5x+1)^8
sqrt(5x+1)8
sqrt5x+18
sqrt(5x+1)⁸
sqrt5x+1^8
Similar expressions
sqrt(5x-1)^8

The derivative of the function/ sqrt(5x+1)^8

Derivative of sqrt(5x+1)^8

The solution

You have entered [src]

           8
  _________ 
\/ 5*x + 1

$$\left(\sqrt{5 x + 1}\right)^{8}$$

(sqrt(5*x + 1))^8

Detail solution

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

$20 \left(5 x + 1\right)^{3}$
Now simplify:

$20 \left(5 x + 1\right)^{3}$

The answer is:

$20 \left(5 x + 1\right)^{3}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

            4
20*(5*x + 1) 
-------------
   5*x + 1

$$\frac{20 \left(5 x + 1\right)^{4}}{5 x + 1}$$

Simplify

The second derivative [src]

             2
300*(1 + 5*x)

$$300 \left(5 x + 1\right)^{2}$$

Simplify

The third derivative [src]

3000*(1 + 5*x)

$$3000 \left(5 x + 1\right)$$

Simplify