What is the derivative of cos(6x)?
Understand the Problem
The question is asking for the derivative of the cosine function with respect to 6x. To solve this, we will apply the chain rule, which involves differentiating the outer function and then multiplying by the derivative of the inner function.
Answer
The derivative is $-6\sin(6x)$.
Answer for screen readers
The derivative of the cosine function with respect to $6x$ is $-6\sin(6x)$.
Steps to Solve
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Identify the function and the inner function We have the function $f(x) = \cos(6x)$. Here, the outer function is $\cos(u)$ and the inner function is $u = 6x$.
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Differentiate the outer function The derivative of $\cos(u)$ with respect to $u$ is $-\sin(u)$.
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Differentiate the inner function Now, we differentiate the inner function $u = 6x$. The derivative of $6x$ with respect to $x$ is $6$.
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Apply the chain rule Using the chain rule, we multiply the derivative of the outer function by the derivative of the inner function: $$ f'(x) = -\sin(6x) \cdot 6 $$
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Simplify the expression This simplifies to: $$ f'(x) = -6\sin(6x) $$
The derivative of the cosine function with respect to $6x$ is $-6\sin(6x)$.
More Information
The derivative $-6\sin(6x)$ tells us how the function $\cos(6x)$ changes as $x$ changes. In practical terms, this means if we know the angle in radians that $6x$ represents, we can find the slope of the cosine function at that point.
Tips
- Forgetting to multiply by the derivative of the inner function. Always remember to apply the chain rule correctly.
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