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The derivative of the function/ sin^2(3x)

Derivative of sin^2(3x)

The solution

You have entered [src]

   2     
sin (3*x)

$$\sin^{2}{\left(3 x \right)}$$

d /   2     \
--\sin (3*x)/
dx

$$\frac{d}{d x} \sin^{2}{\left(3 x \right)}$$

Transform

Detail solution

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

$6 \sin{\left(3 x \right)} \cos{\left(3 x \right)}$
Now simplify:

$3 \sin{\left(6 x \right)}$

The answer is:

$3 \sin{\left(6 x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

6*cos(3*x)*sin(3*x)

$$6 \sin{\left(3 x \right)} \cos{\left(3 x \right)}$$

Simplify

The second derivative [src]

   /   2           2     \
18*\cos (3*x) - sin (3*x)/

$$18 \left(- \sin^{2}{\left(3 x \right)} + \cos^{2}{\left(3 x \right)}\right)$$

Simplify

The third derivative [src]

-216*cos(3*x)*sin(3*x)

$$- 216 \sin{\left(3 x \right)} \cos{\left(3 x \right)}$$

Simplify

The graph

Derivative of sin^2(3x)