Derivative of Cosine
Derivative of the cosine function.
Formula
Variables
Example
The slope of cos x at x = 0 is −sin 0 = 0 (a peak).
How it works
The derivative of cos x tells you the instantaneous rate at which the cosine graph is changing at any x-value. It comes out as -sin x because cosine is a periodic wave whose slopes match the negative of the sine wave at every point; for example, where cosine is at a maximum or minimum, its slope is zero, and where cosine crosses the axis, the slope is steepest. This relationship is one of the basic linked facts in trigonometric calculus.
When to use it
Use this when differentiating expressions that contain cos x, such as in physics, engineering, or calculus problems involving oscillations, circular motion, or wave models. It is especially useful when finding tangent lines, velocity from position functions, or when simplifying derivatives inside larger expressions.
Worked example, step by step
Find the derivative of f(x) = cos x at x = pi/3. Step 1: Differentiate the function: f'(x) = -sin x. Step 2: Substitute x = pi/3: f'(pi/3) = -sin(pi/3). Step 3: Use the exact sine value: sin(pi/3) = sqrt(3)/2. Step 4: Compute the result: f'(pi/3) = -sqrt(3)/2. Final answer: the slope of cos x at x = pi/3 is -sqrt(3)/2.
Common mistakes to avoid
- forgetting the negative sign and writing d/dx(cos x) = sin x
- confusing the derivative with the function itself, especially at special angles
- assuming the rule changes to a different derivative without checking if x is in radians
Did You Know?
Sine and cosine chase each other forever under differentiation, describing all oscillating motion.
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