Derivative of 9cos^3xcospix

The solution

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     3             
9*cos (x)*cos(pi*x)

$$9 \cos^{3}{\left(x \right)} \cos{\left(\pi x \right)}$$

(9*cos(x)^3)*cos(pi*x)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = 9 \cos^{3}{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \cos{\left(x \right)}$ .
  2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
  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:
    
    $- 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}$
  So, the result is: $- 27 \sin{\left(x \right)} \cos^{2}{\left(x \right)}$
$g{\left(x \right)} = \cos{\left(\pi x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \pi x$ .
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 $\frac{d}{d x} \pi x$ :
  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: $\pi$
  The result of the chain rule is:
  
  $- \pi \sin{\left(\pi x \right)}$
The result is: $- 27 \sin{\left(x \right)} \cos^{2}{\left(x \right)} \cos{\left(\pi x \right)} - 9 \pi \sin{\left(\pi x \right)} \cos^{3}{\left(x \right)}$
Now simplify:

$- 9 \left(3 \sin{\left(x \right)} \cos{\left(\pi x \right)} + \pi \sin{\left(\pi x \right)} \cos{\left(x \right)}\right) \cos^{2}{\left(x \right)}$

The answer is:

$- 9 \left(3 \sin{\left(x \right)} \cos{\left(\pi x \right)} + \pi \sin{\left(\pi x \right)} \cos{\left(x \right)}\right) \cos^{2}{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        2                               3             
- 27*cos (x)*cos(pi*x)*sin(x) - 9*pi*cos (x)*sin(pi*x)

$$- 27 \sin{\left(x \right)} \cos^{2}{\left(x \right)} \cos{\left(\pi x \right)} - 9 \pi \sin{\left(\pi x \right)} \cos^{3}{\left(x \right)}$$

Simplify

The second derivative [src]

  /  /     2           2   \               2    2                                            \       
9*\3*\- cos (x) + 2*sin (x)/*cos(pi*x) - pi *cos (x)*cos(pi*x) + 6*pi*cos(x)*sin(x)*sin(pi*x)/*cos(x)

$$9 \left(3 \left(2 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos{\left(\pi x \right)} + 6 \pi \sin{\left(x \right)} \sin{\left(\pi x \right)} \cos{\left(x \right)} - \pi^{2} \cos^{2}{\left(x \right)} \cos{\left(\pi x \right)}\right) \cos{\left(x \right)}$$

Simplify

The third derivative [src]

  /  3    3                  /       2           2   \                         /     2           2   \                        2    2                    \
9*\pi *cos (x)*sin(pi*x) - 3*\- 7*cos (x) + 2*sin (x)/*cos(pi*x)*sin(x) - 9*pi*\- cos (x) + 2*sin (x)/*cos(x)*sin(pi*x) + 9*pi *cos (x)*cos(pi*x)*sin(x)/

$$9 \left(- 3 \left(2 \sin^{2}{\left(x \right)} - 7 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(\pi x \right)} - 9 \pi \left(2 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin{\left(\pi x \right)} \cos{\left(x \right)} + 9 \pi^{2} \sin{\left(x \right)} \cos^{2}{\left(x \right)} \cos{\left(\pi x \right)} + \pi^{3} \sin{\left(\pi x \right)} \cos^{3}{\left(x \right)}\right)$$

Simplify

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Identical expressions

Derivative of 9cos^3xcospix

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Piecewise:

The solution