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y=sin(ctg(9x^ two + five))
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Similar expressions
y=sin(ctg(9x^2-5))

The derivative of the function/ y=sin(ctg(9x^2+5))

Derivative of y=sin(ctg(9x^2+5))

The solution

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   /   /   2    \\
sin\cot\9*x  + 5//

$$\sin{\left(\cot{\left(9 x^{2} + 5 \right)} \right)}$$

sin(cot(9*x^2 + 5))

Detail solution

Let $u = \cot{\left(9 x^{2} + 5 \right)}$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cot{\left(9 x^{2} + 5 \right)}$ :
1. There are multiple ways to do this derivative.
  Method #1
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(9 x^{2} + 5 \right)} = \frac{1}{\tan{\left(9 x^{2} + 5 \right)}}$
  2. Let $u = \tan{\left(9 x^{2} + 5 \right)}$ .
  3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(9 x^{2} + 5 \right)}$ :
    1. Rewrite the function to be differentiated:
      
      $\tan{\left(9 x^{2} + 5 \right)} = \frac{\sin{\left(9 x^{2} + 5 \right)}}{\cos{\left(9 x^{2} + 5 \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(9 x^{2} + 5 \right)}$ and $g{\left(x \right)} = \cos{\left(9 x^{2} + 5 \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      Let $u = 9 x^{2} + 5$ .
      
      The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(9 x^{2} + 5\right)$ :
      
      Differentiate $9 x^{2} + 5$ term by term:
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x^{2}$ goes to $2 x$
      
      So, the result is: $18 x$
      
      The derivative of the constant $5$ is zero.
      
      The result is: $18 x$
      
      The result of the chain rule is:
      
      $18 x \cos{\left(9 x^{2} + 5 \right)}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = 9 x^{2} + 5$ .
      
      The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(9 x^{2} + 5\right)$ :
      
      Differentiate $9 x^{2} + 5$ term by term:
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x^{2}$ goes to $2 x$
      
      So, the result is: $18 x$
      
      The derivative of the constant $5$ is zero.
      
      The result is: $18 x$
      
      The result of the chain rule is:
      
      $- 18 x \sin{\left(9 x^{2} + 5 \right)}$
      
      Now plug in to the quotient rule:
      
      $\frac{18 x \sin^{2}{\left(9 x^{2} + 5 \right)} + 18 x \cos^{2}{\left(9 x^{2} + 5 \right)}}{\cos^{2}{\left(9 x^{2} + 5 \right)}}$
    The result of the chain rule is:
    
    $- \frac{18 x \sin^{2}{\left(9 x^{2} + 5 \right)} + 18 x \cos^{2}{\left(9 x^{2} + 5 \right)}}{\cos^{2}{\left(9 x^{2} + 5 \right)} \tan^{2}{\left(9 x^{2} + 5 \right)}}$
  Method #2
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(9 x^{2} + 5 \right)} = \frac{\cos{\left(9 x^{2} + 5 \right)}}{\sin{\left(9 x^{2} + 5 \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \cos{\left(9 x^{2} + 5 \right)}$ and $g{\left(x \right)} = \sin{\left(9 x^{2} + 5 \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = 9 x^{2} + 5$ .
    2. The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(9 x^{2} + 5\right)$ :
      
      Differentiate $9 x^{2} + 5$ term by term:
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x^{2}$ goes to $2 x$
      
      So, the result is: $18 x$
      
      The derivative of the constant $5$ is zero.
      
      The result is: $18 x$
      
      The result of the chain rule is:
      
      $- 18 x \sin{\left(9 x^{2} + 5 \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = 9 x^{2} + 5$ .
    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 $\frac{d}{d x} \left(9 x^{2} + 5\right)$ :
      
      Differentiate $9 x^{2} + 5$ term by term:
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x^{2}$ goes to $2 x$
      
      So, the result is: $18 x$
      
      The derivative of the constant $5$ is zero.
      
      The result is: $18 x$
      
      The result of the chain rule is:
      
      $18 x \cos{\left(9 x^{2} + 5 \right)}$
    Now plug in to the quotient rule:
    
    $\frac{- 18 x \sin^{2}{\left(9 x^{2} + 5 \right)} - 18 x \cos^{2}{\left(9 x^{2} + 5 \right)}}{\sin^{2}{\left(9 x^{2} + 5 \right)}}$
The result of the chain rule is:

$- \frac{\left(18 x \sin^{2}{\left(9 x^{2} + 5 \right)} + 18 x \cos^{2}{\left(9 x^{2} + 5 \right)}\right) \cos{\left(\cot{\left(9 x^{2} + 5 \right)} \right)}}{\cos^{2}{\left(9 x^{2} + 5 \right)} \tan^{2}{\left(9 x^{2} + 5 \right)}}$
Now simplify:

$- \frac{18 x \cos{\left(\cot{\left(9 x^{2} + 5 \right)} \right)}}{\cos^{2}{\left(9 x^{2} + 5 \right)} \tan^{2}{\left(9 x^{2} + 5 \right)}}$

The answer is:

$- \frac{18 x \cos{\left(\cot{\left(9 x^{2} + 5 \right)} \right)}}{\cos^{2}{\left(9 x^{2} + 5 \right)} \tan^{2}{\left(9 x^{2} + 5 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     /        2/   2    \\    /   /   2    \\
18*x*\-1 - cot \9*x  + 5//*cos\cot\9*x  + 5//

$$18 x \left(- \cot^{2}{\left(9 x^{2} + 5 \right)} - 1\right) \cos{\left(\cot{\left(9 x^{2} + 5 \right)} \right)}$$

Simplify

The second derivative [src]

   /       2/       2\\ /     /   /       2\\       2 /       2/       2\\    /   /       2\\       2    /   /       2\\    /       2\\
18*\1 + cot \5 + 9*x //*\- cos\cot\5 + 9*x // - 18*x *\1 + cot \5 + 9*x //*sin\cot\5 + 9*x // + 36*x *cos\cot\5 + 9*x //*cot\5 + 9*x //

$$18 \left(\cot^{2}{\left(9 x^{2} + 5 \right)} + 1\right) \left(- 18 x^{2} \left(\cot^{2}{\left(9 x^{2} + 5 \right)} + 1\right) \sin{\left(\cot{\left(9 x^{2} + 5 \right)} \right)} + 36 x^{2} \cos{\left(\cot{\left(9 x^{2} + 5 \right)} \right)} \cot{\left(9 x^{2} + 5 \right)} - \cos{\left(\cot{\left(9 x^{2} + 5 \right)} \right)}\right)$$

Simplify

The third derivative [src]

                           /                                                                                                                                                                                                    2                                                                                 \
      /       2/       2\\ |  /       2/       2\\    /   /       2\\        /   /       2\\    /       2\       2    2/       2\    /   /       2\\       2 /       2/       2\\    /   /       2\\      2 /       2/       2\\     /   /       2\\       2 /       2/       2\\    /       2\    /   /       2\\|
972*x*\1 + cot \5 + 9*x //*\- \1 + cot \5 + 9*x //*sin\cot\5 + 9*x // + 2*cos\cot\5 + 9*x //*cot\5 + 9*x / - 24*x *cot \5 + 9*x /*cos\cot\5 + 9*x // - 12*x *\1 + cot \5 + 9*x //*cos\cot\5 + 9*x // + 6*x *\1 + cot \5 + 9*x // *cos\cot\5 + 9*x // + 36*x *\1 + cot \5 + 9*x //*cot\5 + 9*x /*sin\cot\5 + 9*x ///

$$972 x \left(\cot^{2}{\left(9 x^{2} + 5 \right)} + 1\right) \left(6 x^{2} \left(\cot^{2}{\left(9 x^{2} + 5 \right)} + 1\right)^{2} \cos{\left(\cot{\left(9 x^{2} + 5 \right)} \right)} + 36 x^{2} \left(\cot^{2}{\left(9 x^{2} + 5 \right)} + 1\right) \sin{\left(\cot{\left(9 x^{2} + 5 \right)} \right)} \cot{\left(9 x^{2} + 5 \right)} - 12 x^{2} \left(\cot^{2}{\left(9 x^{2} + 5 \right)} + 1\right) \cos{\left(\cot{\left(9 x^{2} + 5 \right)} \right)} - 24 x^{2} \cos{\left(\cot{\left(9 x^{2} + 5 \right)} \right)} \cot^{2}{\left(9 x^{2} + 5 \right)} - \left(\cot^{2}{\left(9 x^{2} + 5 \right)} + 1\right) \sin{\left(\cot{\left(9 x^{2} + 5 \right)} \right)} + 2 \cos{\left(\cot{\left(9 x^{2} + 5 \right)} \right)} \cot{\left(9 x^{2} + 5 \right)}\right)$$

Simplify

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Derivative of y=sin(ctg(9x^2+5))

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The solution

Method #1

Method #2