Derivative of y=ln(sin(ax)+cos(ax))

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

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

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

Detail solution

Let $u = \sin{\left(a x \right)} + \cos{\left(a x \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} \left(\sin{\left(a x \right)} + \cos{\left(a x \right)}\right)$ :
1. Differentiate $\sin{\left(a x \right)} + \cos{\left(a x \right)}$ term by term:
  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:
    
    $a \cos{\left(a x \right)}$
  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: $- a \sin{\left(a x \right)} + a \cos{\left(a x \right)}$
The result of the chain rule is:

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

$\frac{a}{\tan{\left(a x + \frac{\pi}{4} \right)}}$

The answer is:

$\frac{a}{\tan{\left(a x + \frac{\pi}{4} \right)}}$

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of y=ln(sin(ax)+cos(ax))

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