Mister Exam

Other calculators

cos^5x
The derivative of the function/ cos^5x

Derivative of cos^5x

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   5   
cos (x)
cos⁡5(x)\cos^{5}{\left(x \right)} 
d /   5   \
--\cos (x)/
dx
ddxcos⁡5(x)\frac{d}{d x} \cos^{5}{\left(x \right)}
Transform
Detail solution

  1. Let u=cos⁡(x)u = \cos{\left(x \right)}.

  2. Apply the power rule: u5u^{5} goes to 5u45 u^{4}

  3. Then, apply the chain rule. Multiply by ddxcos⁡(x)\frac{d}{d x} \cos{\left(x \right)}:

    1. The derivative of cosine is negative sine:

      ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

    The result of the chain rule is:

    −5sin⁡(x)cos⁡4(x)- 5 \sin{\left(x \right)} \cos^{4}{\left(x \right)}

The answer is:

−5sin⁡(x)cos⁡4(x)- 5 \sin{\left(x \right)} \cos^{4}{\left(x \right)}

The graph
02468-8-6-4-2-10105-5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
      4          
-5*cos (x)*sin(x)
−5sin⁡(x)cos⁡4(x)- 5 \sin{\left(x \right)} \cos^{4}{\left(x \right)}
Simplify
The second derivative [src] 
     3    /     2           2   \
5*cos (x)*\- cos (x) + 4*sin (x)/
5⋅(4sin⁡2(x)−cos⁡2(x))cos⁡3(x)5 \cdot \left(4 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos^{3}{\left(x \right)}
Simplify
The third derivative [src] 
     2    /        2            2   \       
5*cos (x)*\- 12*sin (x) + 13*cos (x)/*sin(x)
5(−12sin⁡2(x)+13cos⁡2(x))sin⁡(x)cos⁡2(x)5 \left(- 12 \sin^{2}{\left(x \right)} + 13 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos^{2}{\left(x \right)}
Simplify
The graph
Derivative of cos^5x
    © Mister Exam – Calculator