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3sqrtx+lnx^3
The derivative of the function/ 3sqrtx+lnx^3

Derivative of 3sqrtx+lnx^3

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
    ___      3   
3*\/ x  + log (x)
3x+log⁡(x)33 \sqrt{x} + \log{\left(x \right)}^{3} 
d /    ___      3   \
--\3*\/ x  + log (x)/
dx
ddx(3x+log⁡(x)3)\frac{d}{d x} \left(3 \sqrt{x} + \log{\left(x \right)}^{3}\right)
Transform
Detail solution

  1. Differentiate 3x+log⁡(x)33 \sqrt{x} + \log{\left(x \right)}^{3} 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\sqrt{x} goes to 12x\frac{1}{2 \sqrt{x}}

      So, the result is: 32x\frac{3}{2 \sqrt{x}}

    2. Let u=log⁡(x)u = \log{\left(x \right)}.

    3. Apply the power rule: u3u^{3} goes to 3u23 u^{2}

    4. Then, apply the chain rule. Multiply by ddxlog⁡(x)\frac{d}{d x} \log{\left(x \right)}:

      1. The derivative of log⁡(x)\log{\left(x \right)} is 1x\frac{1}{x}.

      The result of the chain rule is:

      3log⁡(x)2x\frac{3 \log{\left(x \right)}^{2}}{x}

    The result is: 3log⁡(x)2x+32x\frac{3 \log{\left(x \right)}^{2}}{x} + \frac{3}{2 \sqrt{x}}

The answer is:

3log⁡(x)2x+32x\frac{3 \log{\left(x \right)}^{2}}{x} + \frac{3}{2 \sqrt{x}}

The graph
02468-8-6-4-2-1010-200200
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
               2   
   3      3*log (x)
------- + ---------
    ___       x    
2*\/ x
3log⁡(x)2x+32x\frac{3 \log{\left(x \right)}^{2}}{x} + \frac{3}{2 \sqrt{x}}
Simplify
The second derivative [src] 
  /              2              \
  |    1      log (x)   2*log(x)|
3*|- ------ - ------- + --------|
  |     3/2       2         2   |
  \  4*x         x         x    /
3(−log⁡(x)2x2+2log⁡(x)x2−14x32)3 \left(- \frac{\log{\left(x \right)}^{2}}{x^{2}} + \frac{2 \log{\left(x \right)}}{x^{2}} - \frac{1}{4 x^{\frac{3}{2}}}\right)
Simplify
The third derivative [src] 
  /                              2   \
  |2      3      6*log(x)   2*log (x)|
3*|-- + ------ - -------- + ---------|
  | 3      5/2       3           3   |
  \x    8*x         x           x    /
3⋅(2log⁡(x)2x3−6log⁡(x)x3+2x3+38x52)3 \cdot \left(\frac{2 \log{\left(x \right)}^{2}}{x^{3}} - \frac{6 \log{\left(x \right)}}{x^{3}} + \frac{2}{x^{3}} + \frac{3}{8 x^{\frac{5}{2}}}\right)
Simplify
The graph
Derivative of 3sqrtx+lnx^3
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