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

  1. 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.

  1. 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 $f(g) = \sin(g)$.

  1. 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 $g'(x) = 6$.
  2. Combine the derivatives

Now we combine the derivatives using the chain rule:

$$ f'(x) = \cos(6x) \cdot 6 $$

  1. 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.

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