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

Derivative of sqrt(5*x+1)

The solution

You have entered [src]

  _________
\/ 5*x + 1

$$\sqrt{5 x + 1}$$

sqrt(5*x + 1)

Detail solution

Let $u = 5 x + 1$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x + 1\right)$ :
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:

$\frac{5}{2 \sqrt{5 x + 1}}$
Now simplify:

$\frac{5}{2 \sqrt{5 x + 1}}$

The answer is:

$\frac{5}{2 \sqrt{5 x + 1}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      5      
-------------
    _________
2*\/ 5*x + 1

$$\frac{5}{2 \sqrt{5 x + 1}}$$

Simplify

The second derivative [src]

     -25      
--------------
           3/2
4*(1 + 5*x)

$$- \frac{25}{4 \left(5 x + 1\right)^{\frac{3}{2}}}$$

Simplify

The third derivative [src]

     375      
--------------
           5/2
8*(1 + 5*x)

$$\frac{375}{8 \left(5 x + 1\right)^{\frac{5}{2}}}$$

Simplify

The graph

Derivative of sqrt(5*x+1)