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

  1. Differentiate the constant term 1

The derivative of any constant is 0.

$$ \frac{d}{dx}[1] = 0 $$

  1. Differentiate the sine function sin(x)

The derivative of $\sin(x)$ is $\cos(x)$.

$$ \frac{d}{dx} [\sin(x)] = \cos(x) $$

  1. 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 $0$.

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