What is the derivative of sin(6x)?
Understand the Problem
The question is asking for the derivative of the function sin(6x). To solve it, we will apply the chain rule of differentiation.
Answer
The derivative of the function $\sin(6x)$ is $f'(x) = 6 \cos(6x)$.
Answer for screen readers
The derivative of the function $\sin(6x)$ is $f'(x) = 6 \cos(6x)$.
Steps to Solve
- Identify the outer and inner functions
In the function $f(x) = \sin(6x)$, the outer function is $\sin(u)$ where $u = 6x$, and the inner function is $u = 6x$.
- Differentiate the outer function
The derivative of the outer function $\sin(u)$ with respect to $u$ is $\cos(u)$. So,
$$ \frac{d}{du}[\sin(u)] = \cos(u) $$
- Differentiate the inner function
Now, differentiate the inner function $u = 6x$ with respect to $x$. This gives:
$$ \frac{du}{dx} = 6 $$
- Apply the chain rule
According to the chain rule, we multiply the derivative of the outer function by the derivative of the inner function:
$$ \frac{df}{dx} = \frac{d}{du}[\sin(u)] \cdot \frac{du}{dx} $$
Substituting in what we found:
$$ \frac{df}{dx} = \cos(6x) \cdot 6 $$
- Write the final derivative
Thus, the derivative of $f(x) = \sin(6x)$ is:
$$ f'(x) = 6 \cos(6x) $$
The derivative of the function $\sin(6x)$ is $f'(x) = 6 \cos(6x)$.
More Information
The chain rule is crucial in calculus for differentiating composite functions. The cosine function oscillates between -1 and 1, and the factor of 6 scales the amplitude of the rate of change of the sine function.
Tips
- Forgetting to apply the chain rule when differentiating composite functions.
- Mixing up the derivatives of sine and cosine. Remember that the derivative of $\sin(u)$ is $\cos(u)$ and not the other way around.
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