What is the derivative of cos x?
Understand the Problem
The question is asking for the mathematical derivative of the cosine function with respect to x. The derivative of cos(x) can be approached using differentiation rules from calculus.
Answer
The derivative of $\cos(x)$ is $-\sin(x)$.
Answer for screen readers
The derivative of $\cos(x)$ with respect to $x$ is $-\sin(x)$.
Steps to Solve
- Identify the function to differentiate
We will differentiate the function $f(x) = \cos(x)$ with respect to $x$.
- Apply the differentiation rule for cosine
The derivative of the cosine function is given by the rule: $$ \frac{d}{dx}[\cos(x)] = -\sin(x) $$
- State the final result
Using the differentiation rule, we can state the derivative of the cosine function with respect to $x$ as: $$ f'(x) = -\sin(x) $$
The derivative of $\cos(x)$ with respect to $x$ is $-\sin(x)$.
More Information
The derivative of the cosine function is important in various fields, including physics, engineering, and economics, as it helps to understand rates of change and motion. The negative sign indicates that as the cosine function increases, the sine function decreases, showing the relationship between these two fundamental trigonometric functions.
Tips
- A common mistake is confusing the derivative of cosine with the derivative of sine. Remember that the derivative of $\sin(x)$ is $\cos(x)$ and the derivative of $\cos(x)$ is $-\sin(x)$.
