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Similar expressions
ln^3(x)-2^√x

The derivative of the function/ ln^3(x)+2^√x

Derivative of ln^3(x)+2^√x

The solution

You have entered [src]

             ___
   3       \/ x 
log (x) + 2

$$2^{\sqrt{x}} + \log{\left(x \right)}^{3}$$

log(x)^3 + 2^(sqrt(x))

Detail solution

Differentiate $2^{\sqrt{x}} + \log{\left(x \right)}^{3}$ term by term:
1. Let $u = \log{\left(x \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result of the chain rule is:
  
  $\frac{3 \log{\left(x \right)}^{2}}{x}$
4. Let $u = \sqrt{x}$ .
5. $\frac{d}{d u} 2^{u} = 2^{u} \log{\left(2 \right)}$
6. 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:
  
  $\frac{2^{\sqrt{x}} \log{\left(2 \right)}}{2 \sqrt{x}}$
The result is: $\frac{2^{\sqrt{x}} \log{\left(2 \right)}}{2 \sqrt{x}} + \frac{3 \log{\left(x \right)}^{2}}{x}$

The answer is:

$\frac{2^{\sqrt{x}} \log{\left(2 \right)}}{2 \sqrt{x}} + \frac{3 \log{\left(x \right)}^{2}}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

               ___       
     2       \/ x        
3*log (x)   2     *log(2)
--------- + -------------
    x              ___   
               2*\/ x

$$\frac{2^{\sqrt{x}} \log{\left(2 \right)}}{2 \sqrt{x}} + \frac{3 \log{\left(x \right)}^{2}}{x}$$

Simplify

The second derivative [src]

                            ___             ___        
       2                  \/ x            \/ x     2   
  3*log (x)   6*log(x)   2     *log(2)   2     *log (2)
- --------- + -------- - ------------- + --------------
       2          2             3/2           4*x      
      x          x           4*x

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

Simplify

The third derivative [src]

                                  ___              ___                ___       
                      2         \/ x     2       \/ x     3         \/ x        
6    18*log(x)   6*log (x)   3*2     *log (2)   2     *log (2)   3*2     *log(2)
-- - --------- + --------- - ---------------- + -------------- + ---------------
 3        3           3               2                3/2               5/2    
x        x           x             8*x              8*x               8*x

$$- \frac{3 \cdot 2^{\sqrt{x}} \log{\left(2 \right)}^{2}}{8 x^{2}} + \frac{2^{\sqrt{x}} \log{\left(2 \right)}^{3}}{8 x^{\frac{3}{2}}} + \frac{3 \cdot 2^{\sqrt{x}} \log{\left(2 \right)}}{8 x^{\frac{5}{2}}} + \frac{6 \log{\left(x \right)}^{2}}{x^{3}} - \frac{18 \log{\left(x \right)}}{x^{3}} + \frac{6}{x^{3}}$$

Simplify

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Derivative of ln^3(x)+2^√x

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The solution