What is the derivative of cos(6x)?
Understand the Problem
The question is asking for the derivative of the function cos(6x). To find this, we will apply the chain rule of differentiation.
Answer
The derivative of $y = \cos(6x)$ is $ \frac{dy}{dx} = -6\sin(6x) $.
Answer for screen readers
The derivative of the function $y = \cos(6x)$ is:
$$ \frac{dy}{dx} = -6\sin(6x) $$
Steps to Solve
- Identify the outer and inner functions
In the function $y = \cos(6x)$, the outer function is $y = \cos(u)$ where $u = 6x$ is the inner function.
- Differentiate the outer function
The derivative of $\cos(u)$ with respect to $u$ is:
$$ \frac{dy}{du} = -\sin(u) $$
- Differentiate the inner function
Now, find the derivative of the inner function $u = 6x$ with respect to $x$:
$$ \frac{du}{dx} = 6 $$
- Apply the chain rule
Using the chain rule, which states that $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$, we can combine the derivatives:
$$ \frac{dy}{dx} = -\sin(6x) \cdot 6 $$
- Simplify the expression
Thus, the final derivative is:
$$ \frac{dy}{dx} = -6\sin(6x) $$
The derivative of the function $y = \cos(6x)$ is:
$$ \frac{dy}{dx} = -6\sin(6x) $$
More Information
The derivative tells us how the function changes at any point $x$. In this case, the result $-6\sin(6x)$ indicates that the slope of the tangent to the curve at any point depends on the values of $\sin(6x)$. This uses the chain rule, which is essential for differentiating composite functions.
Tips
Some common mistakes include:
- Forgetting to apply the chain rule: It's important to recognize both the outer function and the inner function when differentiating.
- Miscalculating the derivative of the inner function. Always be careful to differentiate correctly.
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