Derivative of (log3(5x))+(lnsinx)

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

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

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

Detail solution

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

$\frac{1}{\tan{\left(x \right)}} + \frac{1}{x \log{\left(3 \right)}}$

The answer is:

$\frac{1}{\tan{\left(x \right)}} + \frac{1}{x \log{\left(3 \right)}}$

The graph

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The first derivative [src]

   1       cos(x)
-------- + ------
x*log(3)   sin(x)

$$\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{1}{x \log{\left(3 \right)}}$$

Simplify

The second derivative [src]

 /                   2   \
 |        1       cos (x)|
-|1 + --------- + -------|
 |     2             2   |
 \    x *log(3)   sin (x)/

$$- (1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{1}{x^{2} \log{\left(3 \right)}})$$

Simplify

The third derivative [src]

  /               3            \
  |    1       cos (x)   cos(x)|
2*|--------- + ------- + ------|
  | 3             3      sin(x)|
  \x *log(3)   sin (x)         /

$$2 \left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{\cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} + \frac{1}{x^{3} \log{\left(3 \right)}}\right)$$

Simplify

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Derivative of (log3(5x))+(lnsinx)

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