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Derivative of:
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Identical expressions
ax+b/x^(one / two)
ax plus b divide by x to the power of (1 divide by 2)
ax plus b divide by x to the power of (one divide by two)
ax+b/x(1/2)
ax+b/x1/2
ax+b/x^1/2
ax+b divide by x^(1 divide by 2)
Similar expressions
ax-b/x^(1/2)

The derivative of the function/ ax+b/x^(1/2)

Derivative of ax+b/x^(1/2)

The solution

You have entered [src]

        b  
a*x + -----
        ___
      \/ x

$$a x + \frac{b}{\sqrt{x}}$$

a*x + b/sqrt(x)

Detail solution

Differentiate $a x + \frac{b}{\sqrt{x}}$ 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: $x$ goes to $1$
  So, the result is: $a$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \sqrt{x}$ .
  2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{x}$ :
    1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
    The result of the chain rule is:
    
    $- \frac{1}{2 x^{\frac{3}{2}}}$
  So, the result is: $- \frac{b}{2 x^{\frac{3}{2}}}$
The result is: $a - \frac{b}{2 x^{\frac{3}{2}}}$

The answer is:

$a - \frac{b}{2 x^{\frac{3}{2}}}$

The first derivative [src]

      b   
a - ------
       3/2
    2*x

$$a - \frac{b}{2 x^{\frac{3}{2}}}$$

Simplify

The second derivative [src]

 3*b  
------
   5/2
4*x

$$\frac{3 b}{4 x^{\frac{5}{2}}}$$

Simplify

The third derivative [src]

-15*b 
------
   7/2
8*x

$$- \frac{15 b}{8 x^{\frac{7}{2}}}$$

Simplify