Derivative of lnsin(ax)+cos(ax)

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log(sin(a*x)) + cos(a*x)

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

d                           
--(log(sin(a*x)) + cos(a*x))
dx

$$\frac{\partial}{\partial x} \left(\log{\left(\sin{\left(a x \right)} \right)} + \cos{\left(a x \right)}\right)$$

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Detail solution

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

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

The answer is:

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

The first derivative [src]

              a*cos(a*x)
-a*sin(a*x) + ----------
               sin(a*x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of lnsin(ax)+cos(ax)

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