What is the derivative of cos(6x)?
Understand the Problem
The question is asking for the derivative of the function cos(6x). To find the derivative, we will use the chain rule, which states that the derivative of cos(u) is -sin(u) multiplied by the derivative of u.
Answer
$$ \frac{dy}{dx} = -6\sin(6x) $$
Answer for screen readers
$$ \frac{dy}{dx} = -6\sin(6x) $$
Steps to Solve
- Identify the outer and inner functions
In this case, the outer function is $y = \cos(u)$ and the inner function is $u = 6x$.
- Differentiate the outer function
Using the chain rule, the derivative of the outer function $y = \cos(u)$ is: $$ \frac{dy}{du} = -\sin(u) $$
- Differentiate the inner function
Now, we take the derivative of the inner function $u = 6x$: $$ \frac{du}{dx} = 6 $$
- Apply the chain rule
Now we multiply the derivatives from steps 2 and 3: $$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = -\sin(6x) \cdot 6 $$
- Write the final derivative
Thus, the derivative of the function $y = \cos(6x)$ is: $$ \frac{dy}{dx} = -6\sin(6x) $$
$$ \frac{dy}{dx} = -6\sin(6x) $$
More Information
The derivative of a trigonometric function such as cosine involves not only the standard derivative of cosine but also accounts for any inner function through the chain rule. Here, the presence of the $6x$ inside the cosine function leads to a multiplication factor of $6$ in the final derivative.
Tips
- Forgetting to apply the chain rule can lead to incorrect results. Always remember to differentiate both the outer and inner functions.
- Miscalculating the derivative of the inner function. Ensure to correctly differentiate $6x$ as $6$, not $1$.
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