Mister Exam

Derivative of y=sin5x²

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   2     
sin (5*x)
sin2(5x)\sin^{2}{\left(5 x \right)}
d /   2     \
--\sin (5*x)/
dx           
ddxsin2(5x)\frac{d}{d x} \sin^{2}{\left(5 x \right)}
Detail solution
  1. Let u=sin(5x)u = \sin{\left(5 x \right)}.

  2. Apply the power rule: u2u^{2} goes to 2u2 u

  3. Then, apply the chain rule. Multiply by ddxsin(5x)\frac{d}{d x} \sin{\left(5 x \right)}:

    1. Let u=5xu = 5 x.

    2. The derivative of sine is cosine:

      ddusin(u)=cos(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

    3. Then, apply the chain rule. Multiply by ddx5x\frac{d}{d x} 5 x:

      1. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: xx goes to 11

        So, the result is: 55

      The result of the chain rule is:

      5cos(5x)5 \cos{\left(5 x \right)}

    The result of the chain rule is:

    10sin(5x)cos(5x)10 \sin{\left(5 x \right)} \cos{\left(5 x \right)}

  4. Now simplify:

    5sin(10x)5 \sin{\left(10 x \right)}


The answer is:

5sin(10x)5 \sin{\left(10 x \right)}

The graph
02468-8-6-4-2-1010-1010
The first derivative [src]
10*cos(5*x)*sin(5*x)
10sin(5x)cos(5x)10 \sin{\left(5 x \right)} \cos{\left(5 x \right)}
The second derivative [src]
   /   2           2     \
50*\cos (5*x) - sin (5*x)/
50(sin2(5x)+cos2(5x))50 \left(- \sin^{2}{\left(5 x \right)} + \cos^{2}{\left(5 x \right)}\right)
The third derivative [src]
-1000*cos(5*x)*sin(5*x)
1000sin(5x)cos(5x)- 1000 \sin{\left(5 x \right)} \cos{\left(5 x \right)}
The graph
Derivative of y=sin5x²