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Derivative of cos^3(5x-3)

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   3         
cos (5*x - 3)

$$\cos^{3}{\left(5 x - 3 \right)}$$

cos(5*x - 3)^3

Detail solution

Let $u = \cos{\left(5 x - 3 \right)}$ .
Apply the power rule: $u^{3}$ goes to $3 u^{2}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(5 x - 3 \right)}$ :
1. Let $u = 5 x - 3$ .
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 $\frac{d}{d x} \left(5 x - 3\right)$ :
  1. Differentiate $5 x - 3$ term by term:
    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: $5$
    2. The derivative of the constant $-3$ is zero.
    The result is: $5$
  The result of the chain rule is:
  
  $- 5 \sin{\left(5 x - 3 \right)}$
The result of the chain rule is:

$- 15 \sin{\left(5 x - 3 \right)} \cos^{2}{\left(5 x - 3 \right)}$
Now simplify:

$- 15 \sin{\left(5 x - 3 \right)} \cos^{2}{\left(5 x - 3 \right)}$

The answer is:

$- 15 \sin{\left(5 x - 3 \right)} \cos^{2}{\left(5 x - 3 \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2                      
-15*cos (5*x - 3)*sin(5*x - 3)

$$- 15 \sin{\left(5 x - 3 \right)} \cos^{2}{\left(5 x - 3 \right)}$$

Simplify

The second derivative [src]

   /     2                  2          \              
75*\- cos (-3 + 5*x) + 2*sin (-3 + 5*x)/*cos(-3 + 5*x)

$$75 \left(2 \sin^{2}{\left(5 x - 3 \right)} - \cos^{2}{\left(5 x - 3 \right)}\right) \cos{\left(5 x - 3 \right)}$$

Simplify

The third derivative [src]

    /       2                  2          \              
375*\- 2*sin (-3 + 5*x) + 7*cos (-3 + 5*x)/*sin(-3 + 5*x)

$$375 \left(- 2 \sin^{2}{\left(5 x - 3 \right)} + 7 \cos^{2}{\left(5 x - 3 \right)}\right) \sin{\left(5 x - 3 \right)}$$

Simplify