Derivative of ln7sinx+5x

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log(7*sin(x)) + 5*x

$$5 x + \log{\left(7 \sin{\left(x \right)} \right)}$$

log(7*sin(x)) + 5*x

Detail solution

Differentiate $5 x + \log{\left(7 \sin{\left(x \right)} \right)}$ term by term:
1. Let $u = 7 \sin{\left(x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 7 \sin{\left(x \right)}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    So, the result is: $7 \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
4. 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 is: $5 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
Now simplify:

$5 + \frac{1}{\tan{\left(x \right)}}$

The answer is:

$5 + \frac{1}{\tan{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    cos(x)
5 + ------
    sin(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of ln7sinx+5x

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The solution