Derivative of y=log35x+lnsinx

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log(35*x) + log(sin(x))

$$\log{\left(35 x \right)} + \log{\left(\sin{\left(x \right)} \right)}$$

log(35*x) + log(sin(x))

Detail solution

Differentiate $\log{\left(35 x \right)} + \log{\left(\sin{\left(x \right)} \right)}$ term by term:
1. Let $u = 35 x$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 35 x$ :
  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: $35$
  The result of the chain rule is:
  
  $\frac{1}{x}$
4. Let $u = \sin{\left(x \right)}$ .
5. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
6. 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: $\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{1}{x}$
Now simplify:

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

The answer is:

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

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The first derivative [src]

1   cos(x)
- + ------
x   sin(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of y=log35x+lnsinx

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