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The derivative of the function/ sin^2ax

Derivative of sin^2ax

The solution

You have entered [src]

   2     
sin (a*x)

$$\sin^{2}{\left(a x \right)}$$

d /   2     \
--\sin (a*x)/
dx

$$\frac{\partial}{\partial x} \sin^{2}{\left(a x \right)}$$

Transform

Detail solution

Let $u = \sin{\left(a x \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} \sin{\left(a x \right)}$ :
1. Let $u = a x$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} a x$ :
  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: $a$
  The result of the chain rule is:
  
  $a \cos{\left(a x \right)}$
The result of the chain rule is:

$2 a \sin{\left(a x \right)} \cos{\left(a x \right)}$
Now simplify:

$a \sin{\left(2 a x \right)}$

The answer is:

$a \sin{\left(2 a x \right)}$

The first derivative [src]

2*a*cos(a*x)*sin(a*x)

$$2 a \sin{\left(a x \right)} \cos{\left(a x \right)}$$

Simplify

The second derivative [src]

   2 /   2           2     \
2*a *\cos (a*x) - sin (a*x)/

$$2 a^{2} \left(- \sin^{2}{\left(a x \right)} + \cos^{2}{\left(a x \right)}\right)$$

Simplify

The third derivative [src]

    3                  
-8*a *cos(a*x)*sin(a*x)

$$- 8 a^{3} \sin{\left(a x \right)} \cos{\left(a x \right)}$$

Simplify