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Derivative of:
Derivative of x^12
Derivative of -e^x
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Identical expressions
sin(cos^ nine (x))
sinus of ( co sinus of e of to the power of 9(x))
sinus of ( co sinus of e of to the power of nine (x))
sin(cos9(x))
sincos9x
sin(cos⁹(x))
sincos^9x

The derivative of the function/ sin(cos^9(x))

Derivative of sin(cos^9(x))

The solution

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   /   9   \
sin\cos (x)/

$$\sin{\left(\cos^{9}{\left(x \right)} \right)}$$

sin(cos(x)^9)

Detail solution

Let $u = \cos^{9}{\left(x \right)}$ .
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 $\frac{d}{d x} \cos^{9}{\left(x \right)}$ :
1. Let $u = \cos{\left(x \right)}$ .
2. Apply the power rule: $u^{9}$ goes to $9 u^{8}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  The result of the chain rule is:
  
  $- 9 \sin{\left(x \right)} \cos^{8}{\left(x \right)}$
The result of the chain rule is:

$- 9 \sin{\left(x \right)} \cos^{8}{\left(x \right)} \cos{\left(\cos^{9}{\left(x \right)} \right)}$

The answer is:

$- 9 \sin{\left(x \right)} \cos^{8}{\left(x \right)} \cos{\left(\cos^{9}{\left(x \right)} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      8       /   9   \       
-9*cos (x)*cos\cos (x)/*sin(x)

$$- 9 \sin{\left(x \right)} \cos^{8}{\left(x \right)} \cos{\left(\cos^{9}{\left(x \right)} \right)}$$

Simplify

The second derivative [src]

     7    /     2       /   9   \        2       /   9   \        9       2       /   9   \\
9*cos (x)*\- cos (x)*cos\cos (x)/ + 8*sin (x)*cos\cos (x)/ - 9*cos (x)*sin (x)*sin\cos (x)//

$$9 \left(- 9 \sin^{2}{\left(x \right)} \sin{\left(\cos^{9}{\left(x \right)} \right)} \cos^{9}{\left(x \right)} + 8 \sin^{2}{\left(x \right)} \cos{\left(\cos^{9}{\left(x \right)} \right)} - \cos^{2}{\left(x \right)} \cos{\left(\cos^{9}{\left(x \right)} \right)}\right) \cos^{7}{\left(x \right)}$$

Simplify

The third derivative [src]

     6    /        2       /   9   \         11       /   9   \         2       /   9   \         18       2       /   9   \          9       2       /   9   \\       
9*cos (x)*\- 56*sin (x)*cos\cos (x)/ - 27*cos  (x)*sin\cos (x)/ + 25*cos (x)*cos\cos (x)/ + 81*cos  (x)*sin (x)*cos\cos (x)/ + 216*cos (x)*sin (x)*sin\cos (x)//*sin(x)

$$9 \left(216 \sin^{2}{\left(x \right)} \sin{\left(\cos^{9}{\left(x \right)} \right)} \cos^{9}{\left(x \right)} + 81 \sin^{2}{\left(x \right)} \cos^{18}{\left(x \right)} \cos{\left(\cos^{9}{\left(x \right)} \right)} - 56 \sin^{2}{\left(x \right)} \cos{\left(\cos^{9}{\left(x \right)} \right)} - 27 \sin{\left(\cos^{9}{\left(x \right)} \right)} \cos^{11}{\left(x \right)} + 25 \cos^{2}{\left(x \right)} \cos{\left(\cos^{9}{\left(x \right)} \right)}\right) \sin{\left(x \right)} \cos^{6}{\left(x \right)}$$

Simplify

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Derivative of sin(cos^9(x))

The graph:

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The solution