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Derivative of:
Derivative of x^4/log(x)
Derivative of a*cos^3(t)
Derivative of 5ln(3x+5ln(x))
Derivative of 4x^-5
Identical expressions
a*cos^ three (t)
a multiply by co sinus of e of cubed (t)
a multiply by co sinus of e of to the power of three (t)
a*cos3(t)
a*cos3t
a*cos³(t)
a*cos to the power of 3(t)
acos^3(t)
acos3(t)
acos3t
acos^3t

The derivative of the function/ a*cos^3(t)

Derivative of a*cos^3(t)

The solution

You have entered [src]

     3   
a*cos (t)

$$a \cos^{3}{\left(t \right)}$$

d /     3   \
--\a*cos (t)/
dt

$$\frac{\partial}{\partial t} a \cos^{3}{\left(t \right)}$$

Transform

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: $u^{3}$ goes to $3 u^{2}$
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:
  
  $- 3 \sin{\left(t \right)} \cos^{2}{\left(t \right)}$
So, the result is: $- 3 a \sin{\left(t \right)} \cos^{2}{\left(t \right)}$

The answer is:

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

The first derivative [src]

        2          
-3*a*cos (t)*sin(t)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

     /       2           2   \       
-3*a*\- 7*cos (t) + 2*sin (t)/*sin(t)

$$- 3 a \left(2 \sin^{2}{\left(t \right)} - 7 \cos^{2}{\left(t \right)}\right) \sin{\left(t \right)}$$

Simplify