derivative of 1 - sin(x)
Understand the Problem
The question is asking for the derivative of the function 1 - sin(x). To solve this, we will apply basic differentiation rules, particularly the derivative of a constant and the derivative of the sine function.
Answer
$-\cos(x)$
Answer for screen readers
The derivative of $1 - \sin(x)$ is $-\cos(x)$
Steps to Solve
- Differentiate the constant term 1
The derivative of any constant is 0.
$$ \frac{d}{dx}[1] = 0 $$
- Differentiate the sine function sin(x)
The derivative of $\sin(x)$ is $\cos(x)$.
$$ \frac{d}{dx} [\sin(x)] = \cos(x) $$
- Apply the minus sign
Since we are differentiating $1 - \sin(x)$, we account for the negative sign before the sine function.
$$ \frac{d}{dx} [1 - \sin(x)] = 0 - \cos(x) = -\cos(x) $$
The derivative of $1 - \sin(x)$ is $-\cos(x)$
More Information
The sine and cosine functions are fundamental in trigonometry and periodic functions. This derivative shows how these functions change with respect to x.
Tips
A common mistake might be to forget applying the minus sign when differentiating $\sin(x)$, or to erroneously assume that the derivative of $1$ is not $0$.