derivative of sin(2 * 3x)
Understand the Problem
The question is asking for the derivative of the function sin(2 * 3x), which involves applying the chain rule and product rule from calculus.
Answer
The derivative is $f'(x) = 6 \cos(6x)$.
Answer for screen readers
The derivative of the function $f(x) = \sin(2 \cdot 3x)$ is $f'(x) = 6 \cos(6x)$.
Steps to Solve
 Identify the function and rules needed
We have the function $f(x) = \sin(2 \cdot 3x)$, which can be rewritten as $f(x) = \sin(6x)$. To find the derivative, we will need to use the chain rule.
 Apply the chain rule
According to the chain rule, if $f(g(x))$ is a composition of functions, then the derivative is given by $f'(g(x)) \cdot g'(x)$. Here, we recognize that $g(x) = 6x$ and $f(g) = \sin(g)$.

Calculate the derivatives of the outer and inner functions
 The derivative of the outer function $\sin(g)$ is $\cos(g)$.
 The derivative of the inner function $g(x) = 6x$ is $g'(x) = 6$.

Combine the derivatives
Now we combine the derivatives using the chain rule:
$$ f'(x) = \cos(6x) \cdot 6 $$
 Simplify the final expression
Thus, the final derivative is:
$$ f'(x) = 6 \cos(6x) $$
The derivative of the function $f(x) = \sin(2 \cdot 3x)$ is $f'(x) = 6 \cos(6x)$.
More Information
The chain rule is an essential concept in calculus that allows us to differentiate composite functions. The result shows how the rate of change of the sine function is scaled by the factor of 6, due to the linear transformation inside the sine function.
Tips
Common mistakes include forgetting to apply the chain rule properly, especially when dealing with compositions of functions. To avoid this, always identify the outer and inner functions clearly before differentiating.