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Derivative of:
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Identical expressions
one / three sin3x+3/5lnx2
1 divide by 3 sinus of 3x plus 3 divide by 5lnx2
one divide by three sinus of 3x plus 3 divide by 5lnx2
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Similar expressions
1/3sin3x-3/5lnx2

The derivative of the function/ 1/3sin3x+3/5lnx2

Derivative of 1/3sin3x+3/5lnx2

The solution

You have entered [src]

sin(3*x)   3*log(x)  
-------- + --------*2
   3          5

$$2 \frac{3 \log{\left(x \right)}}{5} + \frac{\sin{\left(3 x \right)}}{3}$$

sin(3*x)/3 + (3*log(x)/5)*2

Detail solution

Differentiate $2 \frac{3 \log{\left(x \right)}}{5} + \frac{\sin{\left(3 x \right)}}{3}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = 3 x$ .
  2. The derivative of sine is cosine:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
    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: $3$
    The result of the chain rule is:
    
    $3 \cos{\left(3 x \right)}$
  So, the result is: $\cos{\left(3 x \right)}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    So, the result is: $\frac{3}{5 x}$
  So, the result is: $\frac{6}{5 x}$
The result is: $\cos{\left(3 x \right)} + \frac{6}{5 x}$

The answer is:

$\cos{\left(3 x \right)} + \frac{6}{5 x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 6            
--- + cos(3*x)
5*x

$$\cos{\left(3 x \right)} + \frac{6}{5 x}$$

Simplify

The second derivative [src]

   / 2             \
-3*|---- + sin(3*x)|
   |   2           |
   \5*x            /

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

Simplify

The third derivative [src]

  /               4  \
3*|-3*cos(3*x) + ----|
  |                 3|
  \              5*x /

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

Simplify