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Derivative of:
Derivative of x^4*e^x
Derivative of 2/(x+1)
Derivative of (1/x)^x
Derivative of (1-3x)^4
Identical expressions
a*(cos(t))^ five
a multiply by ( co sinus of e of (t)) to the power of 5
a multiply by ( co sinus of e of (t)) to the power of five
a*(cos(t))5
a*cost5
a*(cos(t))⁵
a(cos(t))^5
a(cos(t))5
acost5
acost^5

The derivative of the function/ a*(cos(t))^5

Derivative of a*(cos(t))^5

The solution

You have entered [src]

     5   
a*cos (t)

$$a \cos^{5}{\left(t \right)}$$

a*cos(t)^5

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \cos{\left(t \right)}$ .
2. Apply the power rule: $u^{5}$ goes to $5 u^{4}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d t} \cos{\left(t \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)}$
  The result of the chain rule is:
  
  $- 5 \sin{\left(t \right)} \cos^{4}{\left(t \right)}$
So, the result is: $- 5 a \sin{\left(t \right)} \cos^{4}{\left(t \right)}$

The answer is:

$- 5 a \sin{\left(t \right)} \cos^{4}{\left(t \right)}$

The first derivative [src]

        4          
-5*a*cos (t)*sin(t)

$$- 5 a \sin{\left(t \right)} \cos^{4}{\left(t \right)}$$

Simplify

The second derivative [src]

       3    /     2           2   \
5*a*cos (t)*\- cos (t) + 4*sin (t)/

$$5 a \left(4 \sin^{2}{\left(t \right)} - \cos^{2}{\left(t \right)}\right) \cos^{3}{\left(t \right)}$$

Simplify

The third derivative [src]

        2    /        2            2   \       
-5*a*cos (t)*\- 13*cos (t) + 12*sin (t)/*sin(t)

$$- 5 a \left(12 \sin^{2}{\left(t \right)} - 13 \cos^{2}{\left(t \right)}\right) \sin{\left(t \right)} \cos^{2}{\left(t \right)}$$

Simplify